\documentclass{article} \usepackage{amssymb} \usepackage{amsmath} \usepackage{url} \urlstyle{same} \def\ul{\underline} \def\ov{\overline} \begin{document} \begin{center}{\Large Annotated Examples 5}\end{center} \begin{center}{\large Greuel, Laplagne, Seelich }\end{center} I don't know the origin or reason to consider this, but it is example $I_1$ in Greuel, Laplagne, Seelisch, and is almost type I. (That is, it can be made type I by the simple change of variables $z=y-x$, $w=zx$.) Well, OK, I still don't necessarily know the origin, but have found it in MAGMA's Normalisation description as an example with commentary: \begin{verbatim} > // now try a harder case - a singular affine form of modular curve X1(11) > I := ideal

; > time Js := Normalisation(I: FFMin := false); Time: 0.110 > #Js; 1 > J := Js[1][1]; > Groebner(J); > J; Ideal of Polynomial ring of rank 5 over Rational Field Lexicographical Order Variables: $.1, $.2, $.3, $.4, $.5 Groebner basis: [ $.1*$.3 - $.1 - 6*$.3 + $.4*$.5^2 - 4*$.4*$.5 + 6*$.4 - $.5^5 + $.5^4 + 11*$.5^3 - 16*$.5^2 + 2*$.5 + 6, $.1*$.4 + 2*$.3 - $.4*$.5^2 + 2*$.4*$.5 - 2*$.4 + $.5^4 - 4*$.5^3 + 4*$.5^2 - 2, $.1*$.5 - 2*$.3 + $.4 + $.5^3 - 2*$.5^2 + $.5 + 1, $.2 - $.3 + $.5^3 - $.5^2, $.3^2 + 3*$.3 - 2*$.4*$.5^2 + 4*$.4*$.5 - 4*$.4 - $.5^6 + 2*$.5^5 + $.5^4 - 10*$.5^3 + 10*$.5^2 - 4, $.3*$.4 - $.3 - $.4*$.5^3 + $.4*$.5^2 - $.4*$.5 + $.4 - $.5^4 + 2*$.5^3 - 2*$.5^2 + 1, $.3*$.5 + $.3 - $.4 - $.5^4 + 2*$.5^2 - $.5 - 1, $.4^2 - 2*$.4*$.5^2 + $.4*$.5 + $.4 - $.5^5 ] > time Js := Normalisation(I); Time: 1.110 > J := Js[1][1]; > Groebner(J); > J; Ideal of Polynomial ring of rank 2 over Rational Field Lexicographical Order Variables: $.1, $.2 Groebner basis: [ $.1^2*$.2 + 2*$.1*$.2 + $.1 - $.2^2 + 2*$.2 + 1 ] > // Minimised result is a cubic equation in K[x,y] - as good as we could get! > // This example takes MUCH longer with the general method - even setting > // UseMax := true. \end{verbatim} It seems to be a curve of genus 1, so should have a presentation (as it does a few lines above) with {\em one} (I repeat {\em 1} ) relation in terms of functions of weights 3 and 2. Good luck figuring that out from anything in SINGULAR, Macaulay2, or MAGMA's IntegralClosure. If this is run as given in SINGULAR 3-1-0 in characteristic 0, \begin{verbatim} SINGULAR / A Computer Algebra System for Polynomial Computations / version 3-1-0 0< by: G.-M. Greuel, G. Pfister, H. Schoenemann \ Mar 2009 FB Mathematik der Universitaet, D-67653 Kaiserslautern \ > LIB "normal.lib"; > ring r=0,(x,y),dp; > ideal i=(x-y)*x*(y+x^2)^3-y^3*(x^3+x*y-y^2); > list nor=normal(i); > nor; [1]: [1]: // characteristic : 0 // number of vars : 6 // block 1 : ordering dp // : names T(1) T(2) T(3) T(4) // block 2 : ordering dp // : names x y // block 3 : ordering C [2]: [1]: _[1]=x2y5 _[2]=x4y4 _[3]=x5y3+2x3y4+xy5 _[4]=x7y2-3x3y4-2xy5 _[5]=y6 > def R=nor[1][1]; > setring R; > normap; normap[1]=x normap[2]=y > norid; norid[1]=2*T(1)*x+T(2)*x-T(3)*y+x norid[2]=-T(1)*x+3*T(3)*x-3*T(3)*y+T(4)*x-T(4)*y-x+y norid[3]=-T(1)*y+x^2 norid[4]=T(1)*x^2-T(2)*y norid[5]=T(3)*x^2-2*T(3)*y-T(4)*y norid[6]=T(1)^2-T(2) norid[7]=T(1)*T(2)+T(1)+2*T(2)-T(3)*x norid[8]=T(2)^2-3*T(1)*x-2*T(1)-T(2)*x-3*T(2)-T(3)*x^2+3*T(3)*x-2*x+y norid[9]=T(1)*T(3)-2*T(3)-T(4) norid[10]=T(2)*T(3)-3*T(1)*x-T(1)-T(2)*x-T(2)-T(3)*x^2+2*T(3)+T(4)-x+y norid[11]=T(3)^2-3*T(1)*x-T(2)*x-T(2)-T(3)*x^2-4*T(3)-T(4)-x+y+1 norid[12]=T(1)*T(4)-3*T(1)*x-T(1)-T(2)*x-T(2)-T(3)*x^2+6*T(3)+3*T(4)-x+y norid[13]=T(2)*T(4)-3*T(1)*x^3-T(1)*x^2+9*T(1)*x+4*T(1)-T(2)*x^3+2*T(2)*x+4*T(2)+3*T(3)*x^2-T(3)*x-6*T(3)-3*T(4)-4*x^3-x*y+y^2+3*x-3*y norid[14]=T(3)*T(4)-3*T(1)*x^3-T(1)*x^2+3*T(1)*x+T(1)-T(2)*x^3+3*T(2)+T(3)*x^2-T(3)*x+6*T(3)+T(4)-4*x^3-x*y+y^2+x-y-2 norid[15]=T(4)^2-T(1)*x^5+8*T(1)*x^3+3*T(1)*x^2-5*T(1)*x-6*T(1)+3*T(2)*x^3-T(2)*x^2+3*T(2)*x-10*T(2)-3*T(3)*x^2+6*T(3)*x-6*T(3)+T(4)-3*x^5-x^4-4*x^3*y+12*x^3-x*y^2+y^3+3*x*y-3*y^2-3*x+2*y+4 norid[16]=x^8-x^7*y+3*x^6*y-3*x^5*y^2+3*x^4*y^2-4*x^3*y^3+x^2*y^3-2*x*y^4+y^5 > option(redSB); > ideal j=std(norid);j; j[1]=x^8-x^7*y+3*x^6*y-3*x^5*y^2+3*x^4*y^2-4*x^3*y^3+x^2*y^3-2*x*y^4+y^5 j[2]=6*T(4)*x^3-5*T(4)*x^2*y-T(4)*y^3+6*T(4)*x*y-3*T(4)*y^2-3*x^5-x^4-7*x^3*y+18*x^3+5*x^2*y-3*x*y^2+y^3+18*x*y-12*y^2 j[3]=T(4)*y^4-x^7+3*x^3*y^2+2*x*y^3 j[4]=T(4)*x*y^3-x^7+3*x^6-3*x^5*y+6*x^4*y-4*x^3*y^2+3*x^2*y^2-2*x*y^3+y^4 j[5]=T(4)*x^2*y^2+3*T(4)*y^3-x^7-3*x^5*y+6*x^5-x^4*y-4*x^3*y^2+12*x^3*y-x^2*y^2-x*y^3+y^4+6*x*y^2 j[6]=6*T(3)*y^2+T(4)*x^2*y-T(4)*y^3+3*T(4)*y^2-3*x^5-x^4-7*x^3*y-x^2*y-3*x*y^2+y^3 j[7]=6*T(3)*x*y+T(4)*x^2*y-T(4)*y^3+2*T(4)*x*y+T(4)*y^2-3*x^5-x^4-7*x^3*y-2*x^3-x^2*y-3*x*y^2+y^3-2*x*y+2*y^2 j[8]=T(3)*x^2-2*T(3)*y-T(4)*y j[9]=2*T(2)*y-12*T(3)*y-T(4)*x^2*y+T(4)*y^3-2*T(4)*x^2-T(4)*y^2-6*T(4)*y+3*x^5+x^4+7*x^3*y+2*x^3+x^2*y+3*x*y^2-y^3+2*x^2-2*y^2 j[10]=T(2)*x+6*T(3)*x-7*T(3)*y+2*T(4)*x-2*T(4)*y-x+2*y j[11]=T(1)*y-x^2 j[12]=T(1)*x-3*T(3)*x+3*T(3)*y-T(4)*x+T(4)*y+x-y j[13]=3*T(4)^2-18*T(1)-30*T(2)-81*T(3)*x+180*T(3)*y-18*T(3)+4*T(4)*x^2*y-7*T(4)*y^3+15*T(4)*x^2+T(4)*x*y+5*T(4)*y^2-33*T(4)*x+69*T(4)*y+3*T(4)-21*x^5-7*x^4-49*x^3*y-x^3-4*x^2*y-21*x*y^2+7*y^3-12*x^2+2*x*y+7*y^2+15*x-27*y+12 j[14]=T(3)*T(4)+T(1)+3*T(2)+8*T(3)*x-13*T(3)*y+6*T(3)-T(4)*x^2+3*T(4)*x-5*T(4)*y+T(4)+x^2-2*x+2*y-2 j[15]=T(2)*T(4)+4*T(1)+4*T(2)+14*T(3)*x-13*T(3)*y-6*T(3)-T(4)*x^2+5*T(4)*x-5*T(4)*y-3*T(4)+x^2-4*x+2*y j[16]=T(1)*T(4)-T(1)-T(2)-3*T(3)*x+6*T(3)-T(4)*x+3*T(4)+x j[17]=T(3)^2-T(2)-3*T(3)*x-4*T(3)-T(4)*x-T(4)+x+1 j[18]=T(2)*T(3)-T(1)-T(2)-3*T(3)*x+2*T(3)-T(4)*x+T(4)+x j[19]=T(1)*T(3)-2*T(3)-T(4) j[20]=T(2)^2-2*T(1)-3*T(2)-T(4)*x j[21]=T(1)*T(2)+T(1)+2*T(2)-T(3)*x j[22]=T(1)^2-T(2) \end{verbatim} it is not too hard to figure out that $T(1)=x^2/y$, $T(2)=x^4/y^2$, $T(3)=x*(x^2+y)^2/y^3$ and $T(4)+3T(3)=x(x^2+y)^3/y^4$. MAGMA gets a similar result if $y$ is treated as the independent variable: \begin{verbatim} > FF:=FunctionField(Q); > P:=PolynomialRing(FF); > f:=(x-y)*x*(y+x^2)^3-y^3*(x^3+x*y-y^2); > Ff:=RationalExtensionRepresentation(FunctionField(f)); > C:=CoefficientRing(Ff); > INT:=Integers(C); > IC:=IntegralClosure(INT,Ff); > B:=Basis(IC);for i in [1..#B] do i, B[i]; end for; 1 1 2 X 3 1/Y*X^2 4 1/Y*X^3 5 1/Y^2*X^4 6 1/Y^3*X^5 + 2/Y^2*X^3 + 1/Y*X 7 1/Y^3*X^6 + 3/Y^2*X^4 + 3/Y*X^2 8 1/Y^4*X^7 + 3/Y^3*X^5 + 3/Y^2*X^3 - 1/Y*X^2 + 1/Y*X \end{verbatim} With the change of variables, MAGMA produces a predictable module basis of size 7, with implicit weights maybe 0,11,15,12,9,8,8, though these don't correspond to the implicit weights of the leading monomials \begin{verbatim} F:=Rationals(); P:=PolynomialRing(F,4); f1:=(x-y)*x*(y+x^2)^3-y^3*(x^3+x*y-y^2);1,f1; f2:=y-x-z; f3:=Resultant(f1,f2,y); f4:=z*x-w; f5:=Resultant(f3,f4,x) div z; f5; FF:=FunctionField(F); P:=PolynomialRing(FF); f:=w^7 + 3*w^6*z + w^6 + 3*w^5*z^3 + 6*w^5*z^2 + 9*w^4*z^4 + 4*w^3*z^6 - w^3*z^5 - 3*w^2*z^7 - 3*w*z^9 - z^11;f; Ff:=RationalExtensionRepresentation(FunctionField(f)); C:=CoefficientRing(Ff); INT:=Integers(C); IC:=IntegralClosure(INT,Ff); B:=Basis(IC); for i in [1..#B] do i,B[i]; end for; Genus(Ff); Loading file "greuel1" w^7 + (3*z + 1)*w^6 + (3*z^3 + 6*z^2)*w^5 + 9*z^4*w^4 + (4*z^6 - z^5)*w^3 - 3*z^7*w^2 - 3*z^9*w - z^11 1 1 2 W 3 1/z*W^2 + 1/z*W 4 1/z^3*W^3 + (3*z + 1)/z^3*W^2 + 6/z*W 5 1/z^5*W^4 + (3*z + 1)/z^5*W^3 + (3*z + 6)/z^3*W^2 + 9/z*W 6 1/z^8*W^5 + (-2*z^2 + 3*z + 1)/z^8*W^4 + (3*z^2 - 3*z + 4)/z^6*W^3 + (-4*z + 3)/z^3*W^2 + (-3*z^2 - 1)/z^3*W - 1/z 7 1/z^12*W^6 + (-z^3 - z^2 + 3*z + 1)/z^12*W^5 + (z^4 + 2*z^3 - 2*z^2 - z + 5)/z^10*W^4 + (-3*z^4 + 3*z^2 - 4*z + 4)/z^8*W^3 + (6*z^5 - 2*z^4 + z^3 - 1)/z^7*W^2 + (4*z^4 - z^3 + z - 2)/z^5*W + (-z^2 + z - 1)/z^3 1 \end{verbatim} My qth power algorithm gives a non-minimized ideal of relations: \begin{verbatim} f_2^2+30*f_4+30*f_3+5, f_3^2+30*f_6+3*f_5+30*f_4+19*f_3+19*f_2+11, f_3*f_2+30*f_5+f_4+3*f_3+17, f_4^2+30*f_8+2*f_5+28*f_2, f_4*f_3+30*f_7+3*f_6+6*f_5+29*f_4+19*f_3+13*f_2+30, f_4*f_2+30*f_6+29*f_5+4*f_3+6*f_2+21, f_5^2+30*f_3*f_7+29*f_8+4*f_6+21*f_5+6*f_4+11*f_3+14*f_2+29, f_5*f_4+30*f_2*f_7+2*f_8+29*f_6+23*f_5+30*f_4+8*f_3+18*f_2+11, f_5*f_3+30*f_8+3*f_7+28*f_6+16*f_5+27*f_4+18*f_3+2*f_2+4, f_5*f_2+30*f_7+f_6+4*f_5+25*f_3+21*f_2+20, f_6^2+30*f_5*f_7+3*f_4*f_7+26*f_3*f_7+16*f_2*f_7+8*f_8+f_7+29*f_6+12*f_5+15*f_4+7*f_2+12, f_6*f_5+30*f_4*f_7+3*f_3*f_7+4*f_2*f_7+25*f_8+30*f_7+24*f_6+20*f_5+13*f_4+17*f_3+15*f_2+24, f_6*f_4+30*f_3*f_7+29*f_8+f_7+9*f_6+22*f_5+2*f_4+26*f_3+2*f_2+28, f_6*f_3+30*f_2*f_7+5*f_8+28*f_7+22*f_6+8*f_5+f_4+19*f_3+23*f_2+28, f_6*f_2+30*f_8+f_7+f_6+27*f_5+f_4+6*f_3+9*f_2+26, f_8^2+30*f_2*f_7^2+2*f_8*f_7+28*f_6*f_7+27*f_5*f_7+30*f_4*f_7+6*f_3*f_7+2*f_2*f_7+5*f_8+23*f_7+2*f_6+6*f_5+5*f_4+22*f_3+10*f_2+25, f_8*f_6+30*f_7^2+3*f_6*f_7+4*f_5*f_7+28*f_4*f_7+3*f_3*f_7+14*f_2*f_7+2*f_8+11*f_7+23*f_6+17*f_5+12*f_4+11*f_3+13*f_2+29, f_8*f_5+30*f_6*f_7+27*f_2*f_7+3*f_8+3*f_7+3*f_6+12*f_5+6*f_4+17*f_3+29*f_2+9, f_8*f_4+30*f_5*f_7+f_2*f_7+2*f_8+30*f_6+25*f_5+30*f_4+4*f_3+12*f_2+21, f_8*f_3+30*f_4*f_7+3*f_3*f_7+6*f_2*f_7+21*f_8+22*f_7+28*f_6+26*f_5+26*f_4+15*f_3+24*f_2+18, f_8*f_2+30*f_3*f_7+29*f_2*f_7+2*f_8+3*f_7+f_6+2*f_5+f_4+26*f_3+23*f_2+15 time1= 10.100 \end{verbatim} Minimizing first produces \begin{verbatim} f_15-f_3*f_2^6-12*f_3*f_2^5-2*f_2^6-51*f_3*f_2^4+13*f_2^5-104*f_3*f_2^3+129*f_2^4-108*f_3*f_2^2+351*f_2^3-52*f_3*f_2+425*f_2^2-8*f_3+224*f_2+36, f_14-f_2^7+2*f_3*f_2^5-6*f_2^6+15*f_3*f_2^4-20*f_2^5+44*f_3*f_2^3-68*f_2^4+63*f_3*f_2^2-179*f_2^3+44*f_3*f_2-266*f_2^2+12*f_3-196*f_2-56, f_13-f_3*f_2^5+f_2^6-13*f_3*f_2^4-f_2^5-40*f_3*f_2^3+29*f_2^4-52*f_3*f_2^2+131*f_2^3-40*f_3*f_2+176*f_2^2-16*f_3+140*f_2+64, f_12-f_2^6+3*f_3*f_2^4-3*f_2^5+16*f_3*f_2^3-13*f_2^4+34*f_3*f_2^2-59*f_2^3+35*f_3*f_2-132*f_2^2+14*f_3-148*f_2-64, f_11-f_3*f_2^4-f_2^5-7*f_3*f_2^3-3*f_2^4-18*f_3*f_2^2+11*f_2^3-20*f_3*f_2+57*f_2^2-8*f_3+80*f_2+36, f_10-f_2^5+4*f_3*f_2^3+f_2^4+11*f_3*f_2^2-13*f_2^3+5*f_3*f_2-59*f_2^2-2*f_3-40*f_2+4, f_9-f_3*f_2^3-2*f_2^4-3*f_3*f_2^2-2*f_2^3-2*f_3*f_2+8*f_2^2+8*f_2, f_8-f_2^4+2*f_3*f_2^2-f_2^3+4*f_3*f_2-8*f_2^2+f_3-20*f_2-5, f_7-f_3*f_2^2-2*f_2^3-3*f_3*f_2-2*f_2^2-2*f_3+8*f_2+8, f_3^2-f_2^3+6*f_3*f_2+4*f_2^2-5*f_3-23*f_2+2, f_6-f_2^3+3*f_3*f_2+2*f_2^2-11*f_2-8, f_5-f_3*f_2-f_2^2-2*f_3+9, f_4-f_2^2+f_3-5 \end{verbatim} from which \begin{verbatim} f_3^2-f_2^3+6*f_3*f_2+4*f_2^2-5*f_3-23*f_2+2 \end{verbatim} can be read off. After all a genus computation gives $g=1$. SINGULAR is very quick, as claimed, but produces the following $7+2$ variable $42$ relation answer. Good luck reading any of it, let alone recovering the elliptic curve from it. I guess ``speed kills''. \begin{verbatim} > LIB "normal.lib"; > ring r=0,(w,z),wp(11,7); > ideal i=w^7+3*w^6*z+w^6+3*w^5*z^3+6*w^5*z^2+9*w^4*z^4+4*w^3*z^6-w^3*z^5-3*w^2*z^7-3*w*z^9-z^11; > list nor=normal(i); > nor; [1]:[1]: // characteristic : 0 // number of vars : 9 // block 1 : ordering dp // : names T(1) T(2) T(3) T(4) T(5) T(6) T(7) // block 2 : ordering wp // : names w z // : weights 11 7 // block 3 : ordering C [2]: [1]: _[1]=-72wz10+324w6z2+288w4z5-216w2z8+311w7+550w5z3-216w3z6+262w6z-59w4z4+26w5z2+13w6 _[2]=972w7z-396wz10+1188w6z2+1584w4z5-1188w2z8+1121w7+3034w5z3-1188w3z6+1450w6z-329w4z4+134w5z2+67w6 _[3]=2187w8+117wz10-351w6z2-468w4z5+351w2z8-331w7-896w5z3+351w3z6-428w6z+97w4z4-40w5z2-20w6 _[4]=78732w2z10-236196w7z2-314928w5z5+236196w3z8-236196w8-708588w6z3+236196w4z6-472392w7z+78732w5z4+38808wz10-116424w6z2-155232w4z5+116424w2z8-109957w7-297530w5z3+116424w3z6-142298w6z+32341w4z4-12934w5z2-6467w6 _[5]=36wz10-108w6z2-108w4z5+108w2z8-103w7-206w5z3+108w3z6-98w6z+31w4z4-10w5z2-5w6 _[6]=162w5z4+198wz10-594w6z2-792w4z5+594w2z8-721w7-1514w5z3+594w3z6-722w6z+163w4z4-70w5z2-35w6 _[7]=-9wz10+27w6z2+36w4z5-18w2z8+4w7+26w5z3-27w3z6-10w6z-40w4z4-26w5z2-4w6 _[8]=3w7+6w5z3+4w3z6+6w6z+9w4z4+6w5z2+w6 > def R=nor[1][1]; > setring R; > normap; normap[1]=w normap[2]=z > norid; norid[1]=-324*T(1)*z-126*T(1)-T(2)-8*T(3)-648*T(5)*z+27*T(5)-48*T(6)+5832*w norid[2]=52488*T(1)*w+47628*T(1)*z+12330*T(1)-5832*T(2)*z+4123*T(2)-1768*T(3)+48*T(4)-6345*T(5)+5520*T(6) norid[3]=257580*T(1)*z+399510*T(1)+52488*T(2)*w+25309*T(2)-23328*T(3)*z+11336*T(3)+264*T(4)-105975*T(5)+156720*T(6) norid[4]=-38151*T(1)*z+560880*T(1)+1181*T(2)+26244*T(3)*w+37684*T(3)-729*T(4)*z-39*T(4)-117180*T(5)+213000*T(6) norid[5]=-57348*T(1)*z-35082*T(1)-2291*T(2)-952*T(3)-24*T(4)+52488*T(5)*w+9369*T(5)-11664*T(6)*z-13776*T(6) norid[6]=-66582*T(1)*z-36252*T(1)-729*T(2)*z-1294*T(2)-1664*T(3)-12*T(4)-59049*T(5)*z+8694*T(5)-8748*T(6)*z-14016*T(6)-104976*T(7)*w+236196*z^2 norid[7]=78732*T(1)*z^2+254178*T(1)*z+109602*T(1)-7695*T(2)*z+11939*T(2)+2592*T(3)*z-56*T(3)+132*T(4)-28431*T(5)*z-33669*T(5)-52488*T(6)*w+47628*T(6)*z+44016*T(6) norid[8]=35964*T(1)*z+15858*T(1)-108*T(2)*z+629*T(2)+108*T(3)*z+688*T(3)+6*T(4)+42282*T(5)*z-3861*T(5)+3078*T(6)*z+6144*T(6)+26244*T(7)*z^2+78732*T(7)*w norid[9]=-634567770*T(1)*z-554409162*T(1)+38263752*T(2)*z^3+65938050*T(2)*z^2+20180421*T(2)*z-63929923*T(2)+6928416*T(3)*z^2+50326272*T(3)*z+2525320*T(3)+577368*T(4)*w-370332*T(4)*z-709896*T(4)-71213094*T(5)*z^2-7053075*T(5)*z+173565369*T(5)-229582512*T(6)*w^2+125183880*T(6)*z^2+169116336*T(6)*w+22135356*T(6)*z-223373136*T(6) norid[10]=576301230*T(1)*z+491351652*T(1)-26572050*T(2)*z^2-29333259*T(2)*z+42232058*T(2)-1889568*T(3)*z^2-10839744*T(3)*z+6904480*T(3)-52488*T(4)*w-134136*T(4)*z+457116*T(4)+114791256*T(5)*z^3+15943230*T(5)*z^2-30672675*T(5)*z-140552874*T(5)-83613384*T(6)*z^2-171478296*T(6)*w+48694284*T(6)*z+195017856*T(6)-459165024*T(7)*w^2 norid[11]=-3858717401154*T(1)*z-843314560434*T(1)-41776143984*T(2)*z^2+95793909903*T(2)*z-80021555593*T(2)+37192366944*T(3)*z^4+59538398112*T(3)*z^3-15992673696*T(3)*z^2-194429911356*T(3)*z-7073197568*T(3)+1262703816*T(4)*w^2-809916084*T(4)*w*z+2971818072*T(4)*z^2-328601124*T(4)*w+882165816*T(4)*z-874446474*T(4)-1966331700*T(5)*z^2+613249170777*T(5)*z+248167562373*T(5)-502096953744*T(6)*w^3-113069387160*T(6)*w^2-36044454384*T(6)*z^3-62905608288*T(6)*w*z+11111604624*T(6)*z^2+412117536420*T(6)*w-1134106170426*T(6)*z-336253200672*T(6)+82649704320*T(7)*w^2 norid[12]=-941209264908*T(1)*z-952032991500*T(1)+18255588840*T(2)*z^2+77886428769*T(2)*z-90008025058*T(2)-15152445792*T(3)*z^3-980685792*T(3)*z^2+33976758636*T(3)*z-8194032752*T(3)-57395628*T(4)*w^2-146677716*T(4)*w*z-408146688*T(4)*z^2+227325528*T(4)*w+311952222*T(4)*z-983246088*T(4)-53282274660*T(5)*z^2-15024827781*T(5)*z+279857482110*T(5)+27894275208*T(6)*z^4-74212547004*T(6)*w^2-46146084912*T(6)*z^3-80239087944*T(6)*w*z+103078296360*T(6)*z^2+346418595504*T(6)*w+12449531034*T(6)*z-379534882080*T(6)-502096953744*T(7)*w^3+108477736920*T(7)*w^2 norid[13]=-50200592301626310*T(1)*z-47348968075492896*T(1)+811902928850208*T(2)*z^2+3847866694870689*T(2)*z-4436140472968492*T(2)-725873783180544*T(3)*z^3+25981331362272*T(3)*z^2+1402128524914488*T(3)*z-433111516151792*T(3)+4518872583696*T(4)*z^5+5523066491184*T(4)*w^3-3542572951416*T(4)*w^2*z+7475665755744*T(4)*z^4+12998732246928*T(4)*w*z^2+3746557013328*T(4)*w^2-1952599264560*T(4)*z^3-8959891186656*T(4)*w*z-17361393063084*T(4)*z^2+9858915150984*T(4)*w+16438258266480*T(4)*z-48419906446956*T(4)-2724629196019320*T(5)*z^2+110678525699121*T(5)*z+13881457370656512*T(5)-2196172075676256*T(6)*w^4-3753174729236400*T(6)*w^3-180392277770136*T(6)*w^2*z-2525826620084400*T(6)*w*z^2-4410421019182368*T(6)*w^2-2026906169976432*T(6)*z^3-3530379709412424*T(6)*w*z+5250330612855648*T(6)*z^2+17321168157587640*T(6)*w-856295581636656*T(6)*z-18867757663087968*T(6)-25979500580622048*T(7)*w^3+3247060999862448*T(7)*w^2*z+5371681160266272*T(7)*w^2 norid[14]=T(1)^2-1256160960/479233*T(1)*w*z+239774763876480/229664268289*T(1)*z^2-18672346500/479233*T(1)*w+67023200627876852/6200935243803*T(1)*z+646740226977852904/167425251582681*T(1)+139573440/479233*T(2)*z^2-905244928/1437699*T(2)*w+4923781524259252/2066978414601*T(2)*z-1092366546821084125/1506827264244129*T(2)-527277440/1437699*T(3)*w+4406055581062720/6200935243803*T(3)*z+1208097140388858472/1506827264244129*T(3)+38770400/12939291*T(4)*z-4543896960507904/502275754748043*T(4)+5024643840/479233*T(5)*w*z+3025145088/479233*T(5)*z^2-7372404936/479233*T(5)*w-8421421502399132/688992804867*T(5)*z-58927522435239347/55808417194227*T(5)-1116587520/479233*T(6)*z^2-1794726339029952/229664268289*T(6)*w+6174691199025040/688992804867*T(6)*z+724251987152149280/502275754748043*T(6)+3836396222023680/229664268289*T(7)*z^2+8899423334860800/229664268289*T(7)*w-192*T(7)*z+13232/9*T(7)-27226305792/479233*w*z+5871971995223040/229664268289*z^2+131273180772/479233*w+5256*z+6785 norid[15]=T(1)*T(2)-5426740800/479233*T(1)*w*z+822422696188896/229664268289*T(1)*z^2-80561998005/479233*T(1)*w+179507539039547177/6200935243803*T(1)*z+550334389383450905/334850503165362*T(1)+602971200/479233*T(2)*z^2-3908230216/1437699*T(2)*w+21693736906474813/2066978414601*T(2)*z-21813683248143233419/6027309056976516*T(2)-2277891200/1437699*T(3)*w+18457412577666976/6200935243803*T(3)*z+4642734647748534106/1506827264244129*T(3)+167492000/12939291*T(4)*z-21522035752129960/502275754748043*T(4)+21706963200/479233*T(5)*w*z+13129367616/479233*T(5)*z^2-31778384202/479233*T(5)*w-46047087948270287/688992804867*T(5)*z-971844955197162521/223233668776908*T(5)-4823769600/479233*T(6)*z^2-7601456795057088/229664268289*T(6)*w+25624604167505596/688992804867*T(6)*z+355926764395462496/502275754748043*T(6)+13158763139022336/229664268289*T(7)*z^2+30524845964828160/229664268289*T(7)*w-18336*T(7)*z+124688/9*T(7)-118164308544/479233*w*z+20140747767537408/229664268289*z^2+566198800845/479233*w-36000*z+63215 norid[16]=T(2)^2-21695279760/479233*T(1)*w*z+1793463050814336/229664268289*T(1)*z^2-321343496430/479233*T(1)*w-176763575729439766/6200935243803*T(1)*z-26094774348945789275/167425251582681*T(1)+2410586640/479233*T(2)*z^2-15606873304/1437699*T(2)*w+89017013856049846/2066978414601*T(2)*z-58750283064834329249/3013654528488258*T(2)-9106660640/1437699*T(3)*w+69748357027189456/6200935243803*T(3)*z+11862411427299947764/1506827264244129*T(3)+669607400/12939291*T(4)*z-104647118268711256/502275754748043*T(4)+86781119040/479233*T(5)*w*z+52912404864/479233*T(5)*z^2-126546976908/479233*T(5)*w-264494286103683170/688992804867*T(5)*z-1474481062114020547/111616834388454*T(5)-19284693120/479233*T(6)*z^2-29325552989971920/229664268289*T(6)*w+97765041097376992/688992804867*T(6)*z-28717350417376311676/502275754748043*T(6)+28695408813029376/229664268289*T(7)*z^2+45135317530987392/229664268289*T(7)*w-294384*T(7)*z+1192592/9*T(7)-476211643776/479233*w*z+43921072587414528/229664268289*z^2+2257173090942/479233*w-943308*z+600785 norid[17]=T(1)*T(3)-5498642880/479233*T(1)*w^2*z+1841377965603264/229664268289*T(1)*w*z^2-82122527970/479233*T(1)*w^2+26610785784503370/229664268289*T(1)*w*z-40840567259832/229664268289*T(1)*z^2+108476754007184709/918657073156*T(1)*w+203798349881280767/24803740975212*T(1)*z-8410501951587922627/1339402012661448*T(1)+610960320/479233*T(2)*w*z^2-1323968976/479233*T(2)*w^2+2220620761491282/229664268289*T(2)*w*z-294211476/479233*T(2)*z^2-735673017234523/1377985609734*T(2)*w-39038143394822441/8267913658404*T(2)*z+41031792394797388211/24109236227906064*T(2)-769357440/479233*T(3)*w^2+793634168909952/229664268289*T(3)*w*z+3555785323882292/688992804867*T(3)*w-7633209805075544/6200935243803*T(3)*z-4642558715313162571/3013654528488258*T(3)+6285600/479233*T(4)*w*z-5885322359056/229664268289*T(4)*w-81725410/12939291*T(4)*z+10552663860539534/502275754748043*T(4)+21994571520/479233*T(5)*w^2*z+13017863808/479233*T(5)*w*z^2-32535447876/479233*T(5)*w^2-2281967294824818/229664268289*T(5)*w*z-9498474648/479233*T(5)*z^2+3976548453027684/229664268289*T(5)*w+118958351800216207/2755971219468*T(5)*z+143604411625773325/892934675107632*T(5)-4887682560/479233*T(6)*w*z^2-8419812991063488/229664268289*T(6)*w^2+10308885367903848/229664268289*T(6)*w*z+2353691808/479233*T(6)*z^2+7043568443667332/229664268289*T(6)*w-7478339593434683/688992804867*T(6)*z-1059974294406495100/502275754748043*T(6)+29462047449652224/229664268289*T(7)*w*z^2+68344148360125440/229664268289*T(7)*w^2-653449076157312/229664268289*T(7)*z^2-10445175570611040/229664268289*T(7)*w-32736*T(7)*z+2240/9*T(7)-117160774272/479233*w^2*z+45094486474870272/229664268289*w*z^2+578019809106/479233*w^2+85486271832/479233*w*z-1000166420108736/229664268289*z^2-942121616601/1916932*w-151290*z+860 norid[18]=T(2)*T(3)-30242535840/479233*T(1)*w^2*z+10127578810817952/229664268289*T(1)*w*z^2-451673903835/479233*T(1)*w^2+145205096428194795/229664268289*T(1)*w*z+581926950910008/229664268289*T(1)*z^2+527668473443178597/918657073156*T(1)*w+1694199207335640869/24803740975212*T(1)*z-71570004131900166229/1339402012661448*T(1)+3360281760/479233*T(2)*w*z^2-7281829368/479233*T(2)*w^2+12213414188202051/229664268289*T(2)*w*z-1350553698/479233*T(2)*z^2-11427390319644629/2755971219468*T(2)*w-124220322380279813/8267913658404*T(2)*z+123650455048766954621/24109236227906064*T(2)-4231465920/479233*T(3)*w^2+4364987929004736/229664268289*T(3)*w*z+19072329612914246/688992804867*T(3)*w-29248322579222648/6200935243803*T(3)*z-21375358756891889737/3013654528488258*T(3)+34570800/479233*T(4)*w*z-32369272974808/229664268289*T(4)*w-375153805/12939291*T(4)*z+33315401565213596/502275754748043*T(4)+120970143360/479233*T(5)*w^2*z+71598250944/479233*T(5)*w*z^2-178944963318/479233*T(5)*w^2-7933918575241539/229664268289*T(5)*w*z-71693580420/479233*T(5)*z^2+60165878027096331/918657073156*T(5)*w+255378491100057703/2755971219468*T(5)*z+7103345171608764163/892934675107632*T(5)-26882254080/479233*T(6)*w*z^2-46308971450849184/229664268289*T(6)*w^2+56698869523471164/229664268289*T(6)*w*z+10804429584/479233*T(6)*z^2+37060730096050070/229664268289*T(6)*w-3921778054448807/688992804867*T(6)*z-9744180690680547163/502275754748043*T(6)+162041260973087232/229664268289*T(7)*w*z^2+375892815980689920/229664268289*T(7)*w^2+9310831214560128/229664268289*T(7)*z^2-107876865035574960/229664268289*T(7)*w-308784*T(7)*z+28160/9*T(7)-644384258496/479233*w^2*z+248019675611786496/229664268289*w*z^2+3179108950083/479233*w^2+418856388444/479233*w*z+14251119274459584/229664268289*z^2-4168944884199/1916932*w-1407141*z+12740 norid[19]=T(3)^2+17870589360/479233*T(1)*w^2*z-5984478388210608/229664268289*T(1)*w*z^2+533796431805/958466*T(1)*w^2-169790827920425595/459328536578*T(1)*w*z-500423392266876/229664268289*T(1)*z^2-129015963308249199/459328536578*T(1)*w-2142407986648711295/12401870487606*T(1)*z-95294929985502435067/1339402012661448*T(1)-1985621040/479233*T(2)*w*z^2+4302899172/479233*T(2)*w^2-14434034949693333/459328536578*T(2)*w*z+4319500077/958466*T(2)*z^2+6242755074518677/1837314146312*T(2)*w+600162782540975285/16535827316808*T(2)*z-424205853695254437853/24109236227906064*T(2)+2500411680/479233*T(3)*w^2-2579311048957344/229664268289*T(3)*w*z-3629682024574593/229664268289*T(3)*w+54694653740256661/6200935243803*T(3)*z+19000101580005477323/3013654528488258*T(3)-20428200/479233*T(4)*w*z+19127297666932/229664268289*T(4)*w+2399722265/51757164*T(4)*z-102242987266142479/502275754748043*T(4)-71482357440/479233*T(5)*w^2*z-42308057376/479233*T(5)*w*z^2+105740205597/479233*T(5)*w^2+2115668700976077/459328536578*T(5)*w*z+63756989127/479233*T(5)*z^2-28753527883307175/1837314146312*T(5)*w-2645783813596317889/5511942438936*T(5)*z+26871643336626073741/892934675107632*T(5)+15884968320/479233*T(6)*w*z^2+27364392220956336/229664268289*T(6)*w^2-33503877445687506/229664268289*T(6)*w*z-17278000308/479233*T(6)*z^2-9349330901024083/459328536578*T(6)*w+63277796962119067/688992804867*T(6)*z-15254203158812671561/502275754748043*T(6)-95751654211369728/229664268289*T(7)*w*z^2-222118482170407680/229664268289*T(7)*w^2+472392*T(7)*w*z-8006774276270016/229664268289*T(7)*z^2-163508055533536479/229664268289*T(7)*w-4719*T(7)*z+3200/9*T(7)+380772516384/479233*w^2*z-146557081043328384/229664268289*w*z^2-3757128759189/958466*w^2-205935919722/479233*w*z-12255135184532448/229664268289*z^2+1675322282049/1916932*w-52551/4*z+2000 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norid[32]=T(4)*T(6)-1232875304190067896/229664268289*T(1)*w^2*z+348506253216/479233*T(1)*z^4+336928458809307222790416/110062696284942337*T(1)*w*z^2-17483231211655373055/229664268289*T(1)*w^2+4992090876036/479233*T(1)*z^3+6790082293810927687566591/220125392569884674*T(1)*w*z+65150778183645744/229664268289*T(1)*z^2-10779820793335059969252483/220125392569884674*T(1)*w-83010997229699979209/12401870487606*T(1)*z-2688234980427605522807/669701006330724*T(1)+146264794757088840/229664268289*T(2)*w*z^2-287924115709003554/229664268289*T(2)*w^2+82202191632/479233*T(2)*z^3+582000953077517965020693/110062696284942337*T(2)*w*z+206195191074/479233*T(2)*z^2-273397884104140708699860/110062696284942337*T(2)*w+7992507250063246273/2066978414601*T(2)*z-19005815323725899053769/12054618113953032*T(2)-162191460426956208/229664268289*T(3)*w^2+48762191808/479233*T(3)*z^3+182139583288506854588208/110062696284942337*T(3)*w*z+129761211868471567438724/110062696284942337*T(3)*w+5755549840041142072/6200935243803*T(3)*z+1341545241522825541291/1506827264244129*T(3)+1504781839064700/229664268289*T(4)*w*z-2145993336546548444024/110062696284942337*T(4)*w+57276441965/12939291*T(4)*z-9198284879438543014/502275754748043*T(4)+4764485519512808256/229664268289*T(5)*w^2*z-1394025012864/479233*T(5)*z^4+2628877698124667568/229664268289*T(5)*w*z^2-6719423914598580540/229664268289*T(5)*w^2+1917059797224/479233*T(5)*z^3-596314394909844676939350/110062696284942337*T(5)*w*z+4339591020036/479233*T(5)*z^2-2697629696519061578435973/110062696284942337*T(5)*w-28350602704337623040/688992804867*T(5)*z+209903997058914629249/446467337553816*T(5)-1170118358056710720/229664268289*T(6)*w*z^2-1592000225102309406083928/110062696284942337*T(6)*w^2+1612581895584/479233*T(6)*z^3+2509354940797551888886284/110062696284942337*T(6)*w*z-1649561528592/479233*T(6)*z^2-825891914426170621566200/110062696284942337*T(6)*w+7779083991703819480/688992804867*T(6)*z-823780818364596197665/502275754748043*T(6)+5576100051456/479233*T(7)*w^2*z+5576100051456/479233*T(7)*z^4+5390855340948915564646656/110062696284942337*T(7)*w*z^2+7831411003785050517680064/110062696284942337*T(7)*w^2-7753216211136/479233*T(7)*z^3-8631072657360/479233*T(7)*w*z+1042412450938331904/229664268289*T(7)*z^2+3253907705143089792/229664268289*T(7)*w-2742600*T(7)*z+30372656/9*T(7)-24600673693704337776/229664268289*w^2*z+8534755392768/479233*z^4+8251220614447575272458368/110062696284942337*w*z^2+120505382978179216509/229664268289*w^2-34895172005298/479233*z^3-147705005934909/958466*w*z+1595512133038461312/229664268289*z^2+395976032690079/958466*w-641448*z+15412673 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norid[34]=T(6)^2+586710864/479233*T(1)*z^3-7581237606/479233*T(1)*w*z+5515682136388866/229664268289*T(1)*z^2-534687549513/1916932*T(1)*w+322142115536133497/6200935243803*T(1)*z+2245093345637941579/167425251582681*T(1)+32595048/479233*T(2)*w*z+1245289698/479233*T(2)*z^2-112824811/25446*T(2)*w+78521206502069879/4133956829202*T(2)*z-19486506362471359535/3013654528488258*T(2)+82091232/479233*T(3)*z^2-3355285268/1437699*T(3)*w+33693100038138484/6200935243803*T(3)*z+8559898666787805772/1506827264244129*T(3)+1073088/479233*T(4)*w+307472825/12939291*T(4)*z-39446056294123300/502275754748043*T(4)-2346843456/479233*T(5)*z^3+39481272072/479233*T(5)*w*z+28323473328/479233*T(5)*z^2-54958954320/479233*T(5)*w-173326898140234087/1377985609734*T(5)*z-515797608925571353/111616834388454*T(5)-391140576/479233*T(6)*w*z-6140433024/479233*T(6)*z^2-11736211991410374/229664268289*T(6)*w+46251852525298072/688992804867*T(6)*z+2406532309121146694/502275754748043*T(6)+9387373824/479233*T(7)*w^2+9387373824/479233*T(7)*z^3+21776221440/479233*T(7)*w*z+17554631264314272/229664268289*T(7)*z^2+48756131771383992/229664268289*T(7)*w-15996*T(7)*z+121904/9*T(7)+14368275072/479233*z^3-155963323044/479233*w*z+8290188793759311/229664268289*z^2+3865906353219/1916932*w-20367*z+62177 norid[35]=T(1)*T(7)-35393760/479233*T(1)*w*z-17700721634952/229664268289*T(1)*z^2-2070188265/1916932*T(1)*w-40711534395011735/24803740975212*T(1)*z-949563806801447225/1339402012661448*T(1)+3932640/479233*T(2)*z^2-25299650/1437699*T(2)*w+753393230007713/8267913658404*T(2)*z-1510586685131201183/24109236227906064*T(2)-14856640/1437699*T(3)*w+58267655110808/6200935243803*T(3)*z-12893408940589055/3013654528488258*T(3)+1092400/12939291*T(4)*z-345389521228346/502275754748043*T(4)+141575040/479233*T(5)*w*z+90196560/479233*T(5)*z^2-403775865/958466*T(5)*w-6559703552312515/2755971219468*T(5)*z+137749307497335839/892934675107632*T(5)-31461120/479233*T(6)*z^2-33471740292672/229664268289*T(6)*w+54967033155923/688992804867*T(6)*z-138543335830720355/502275754748043*T(6)-283211546159232/229664268289*T(7)*z^2-656975791009920/229664268289*T(7)*w-24*T(7)*z+799/9*T(7)-811769040/479233*w*z-433482406802496/229664268289*z^2+14494920033/1916932*w+252*z+406 norid[36]=T(2)*T(7)-593282880/479233*T(1)*w*z-550283087576664/229664268289*T(1)*z^2-34602237975/1916932*T(1)*w-1183756213273289297/24803740975212*T(1)*z-26283996830063217287/1339402012661448*T(1)+65920320/479233*T(2)*z^2-423484814/1437699*T(2)*w+14776951387477535/8267913658404*T(2)*z-35974692190002844793/24109236227906064*T(2)-249032320/1437699*T(3)*w+298590012551816/6200935243803*T(3)*z-1063797123040595945/3013654528488258*T(3)+18311200/12939291*T(4)*z-8053844746701494/502275754748043*T(4)+2373131520/479233*T(5)*w*z+1526234544/479233*T(5)*z^2-6734506311/958466*T(5)*w-167921946831182365/2755971219468*T(5)*z+4494761890437912329/892934675107632*T(5)-527362560/479233*T(6)*z^2-389725057835040/229664268289*T(6)*w-286458651421219/688992804867*T(6)*z-3864874883996872283/502275754748043*T(6)-8804529401226624/229664268289*T(7)*z^2-20424176719789440/229664268289*T(7)*w-1320*T(7)*z+7441/9*T(7)-13736110896/479233*w*z-13476175838753472/229664268289*z^2+242101927839/1916932*w-3204*z+3754 norid[37]=T(3)*T(7)-687330360/479233*T(1)*w^2*z+230172245700408/229664268289*T(1)*w*z^2-41061263985/1916932*T(1)*w^2+9698441179415175/918657073156*T(1)*w*z-421450342864812/229664268289*T(1)*z^2-75018359170127757/1837314146312*T(1)*w-1100363740034352851/24803740975212*T(1)*z-24002491287342350951/1339402012661448*T(1)+76370040/479233*T(2)*w*z^2-165496122/479233*T(2)*w^2+1110310380745641/918657073156*T(2)*w*z+172293273/479233*T(2)*z^2-10888238998585309/11023884877872*T(2)*w+29501824778423699/8267913658404*T(2)*z-51994476222087825107/24109236227906064*T(2)-96169680/479233*T(3)*w^2+99204271113744/229664268289*T(3)*w*z+131931773955503/1377985609734*T(3)*w+3475762593465299/6200935243803*T(3)*z+215815088132283098/1506827264244129*T(3)+785700/479233*T(4)*w*z-735665294882/229664268289*T(4)*w+95718485/25878582*T(4)*z-24117901777258351/1004551509496086*T(4)+2749321440/479233*T(5)*w^2*z+1627232976/479233*T(5)*w*z^2-8133861969/958466*T(5)*w^2+13286823203933631/918657073156*T(5)*w*z+4110172236/479233*T(5)*z^2-70060686538169763/3674628292624*T(5)*w-205921801099741963/2755971219468*T(5)*z+5101486792431374195/892934675107632*T(5)-610960320/479233*T(6)*w*z^2-1052476623882936/229664268289*T(6)*w^2+1288610670987981/229664268289*T(6)*w*z-1378346184/479233*T(6)*z^2-1168529367891662/229664268289*T(6)*w+8076974854665763/1377985609734*T(6)*z-14554432208372920955/2009103018992172*T(6)+3682755931206528/229664268289*T(7)*w*z^2+8543018545015680/229664268289*T(7)*w^2-6743205485836992/229664268289*T(7)*z^2-16256343675892017/229664268289*T(7)*w-1905*T(7)*z+100/9*T(7)-14645096784/479233*w^2*z+5636810809358784/229664268289*w*z^2+289009904553/1916932*w^2-36991550124/479233*w*z-10321122084199776/229664268289*z^2+156079567119/479233*w-17613/2*z+40 norid[38]=T(4)*T(7)+18129831984641988/229664268289*T(1)*w^2*z-15841193328/479233*T(1)*z^4+1018459578882056091048/110062696284942337*T(1)*w*z^2+228505111717799745/229664268289*T(1)*w^2-226913221638/479233*T(1)*z^3+296134268263557534059589/440250785139769348*T(1)*w*z+32526474310925352/229664268289*T(1)*z^2+148987206087361898200749/110062696284942337*T(1)*w+68861758252802246311/24803740975212*T(1)*z+1531706156121532757641/1339402012661448*T(1)-2436182587302300/229664268289*T(2)*w*z^2+3959461346628951/229664268289*T(2)*w^2-3736463256/479233*T(2)*z^3-12014816548984509892410/110062696284942337*T(2)*w*z-3292999992/479233*T(2)*z^2+63678938670084537639451/880501570279538696*T(2)*w-777856853954799091/8267913658404*T(2)*z+1989445773851162305711/24109236227906064*T(2)+2068065631462824/229664268289*T(3)*w^2-2216463264/479233*T(3)*z^3-2448100461768277413096/110062696284942337*T(3)*w*z+134988358597065407987/110062696284942337*T(3)*w-1213024238167132/6200935243803*T(3)*z+68008279741735378051/3013654528488258*T(3)-25063606865250/229664268289*T(4)*w*z+71850617929120035224/110062696284942337*T(4)*w-914722220/12939291*T(4)*z+444296438929924756/502275754748043*T(4)-64927705336410528/229664268289*T(5)*w^2*z+63364773312/479233*T(5)*z^4-29743817565952296/229664268289*T(5)*w*z^2+81115310554251039/229664268289*T(5)*w^2-87139081692/479233*T(5)*z^3+336019050668440459904499/220125392569884674*T(5)*w*z-78266314008/479233*T(5)*z^2+191340763120606700689587/880501570279538696*T(5)*w+9504087132617290817/2755971219468*T(5)*z-248965224547010563519/892934675107632*T(5)+19489460698418400/229664268289*T(6)*w*z^2+11888480408477304226068/110062696284942337*T(6)*w^2-73299177072/479233*T(6)*z^3-37259663106626819589060/110062696284942337*T(6)*w*z+26343999936/479233*T(6)*z^2+27658944824730137149394/110062696284942337*T(6)*w+40785608400578153/688992804867*T(6)*z+223970085848902881568/502275754748043*T(6)-253459093248/479233*T(7)*w^2*z-253459093248/479233*T(7)*z^4+16295353262112897456768/110062696284942337*T(7)*w*z^2+250252988875880420094048/110062696284942337*T(7)*w^2+352418918688/479233*T(7)*z^3+364880777112/479233*T(7)*w*z+520423588974805632/229664268289*T(7)*z^2+1191172091049056544/229664268289*T(7)*w+39180*T(7)*z-597521/9*T(7)+310456831301858376/229664268289*w^2*z-387943426944/479233*z^4+24941599477678436445504/110062696284942337*w*z^2-1492405957571142186/229664268289*w^2+1586144182059/479233*z^3+9061844052735/1916932*w*z+796558166377701696/229664268289*z^2-12204512393307/1916932*w-58770*z-301994 norid[39]=T(5)*T(7)+18168300/479233*T(1)*w*z+8692492336200/229664268289*T(1)*z^2+1063256865/1916932*T(1)*w+20121120686857699/24803740975212*T(1)*z+456361630372492789/1339402012661448*T(1)-2018700/479233*T(2)*z^2+12990352/1437699*T(2)*w-383550382786237/8267913658404*T(2)*z+758953262248989547/24109236227906064*T(2)+7626200/1437699*T(3)*w-30970943687140/6200935243803*T(3)*z+5313029092232371/3013654528488258*T(3)-560750/12939291*T(4)*z+173798258556496/502275754748043*T(4)-72673200/479233*T(5)*w*z-46214352/479233*T(5)*z^2+207466713/958466*T(5)*w+3276977410679735/2755971219468*T(5)*z-69434752999115227/892934675107632*T(5)+16149600/479233*T(6)*z^2+17457733400028/229664268289*T(6)*w-30134978863945/688992804867*T(6)*z+66834515330091739/502275754748043*T(6)+139079877379200/229664268289*T(7)*z^2+322628486352000/229664268289*T(7)*w+12*T(7)*z-263/9*T(7)+415929168/479233*w*z+212875078017600/229664268289*z^2-7445683545/1916932*w-72*z-134 norid[40]=T(6)*T(7)-4258170/479233*T(1)*w*z+49720772197152/229664268289*T(1)*z^2-376051125/1916932*T(1)*w+97634068397168269/24803740975212*T(1)*z+2453984934246397735/1339402012661448*T(1)+473130/479233*T(2)*z^2-3809531/1437699*T(2)*w-310915285689997/8267913658404*T(2)*z+1985983424279981917/24109236227906064*T(2)-1787380/1437699*T(3)*w+147724154797202/6200935243803*T(3)*z+176795786395050499/3013654528488258*T(3)+131425/12939291*T(4)*z+417019242957433/502275754748043*T(4)+17032680/479233*T(5)*w*z-7527024/479233*T(5)*z^2-91843569/958466*T(5)*w+10949018392316195/2755971219468*T(5)*z-273434055836603101/892934675107632*T(5)-3785040/479233*T(6)*z^2-41529665356482/229664268289*T(6)*w+264468320992952/688992804867*T(6)*z+697840758579274343/1004551509496086*T(6)+795532355154432/229664268289*T(7)*z^2+1845424402321920/229664268289*T(7)*w+210*T(7)*z-2417/9*T(7)+67743216/479233*w*z+1217638492068096/229664268289*z^2+2856140301/1916932*w-99*z-1226 norid[41]=T(7)^2+2293515/479233*T(1)*w*z-5330028431340/229664268289*T(1)*z^2+142545405/1916932*T(1)*w-2535799111582051/6200935243803*T(1)*z-228891048291682333/1339402012661448*T(1)-254835/479233*T(2)*z^2+3380107/2875398*T(2)*w+1989891159287/4133956829202*T(2)*z-172693829774896657/24109236227906064*T(2)+962710/1437699*T(3)*w-21209614305155/6200935243803*T(3)*z-10285115533590137/1506827264244129*T(3)-141575/25878582*T(4)*z-70423906521929/1004551509496086*T(4)-9174060/479233*T(5)*w*z-4629456/479233*T(5)*z^2+29025639/958466*T(5)*w-530924232800389/1377985609734*T(5)*z+34053493628001841/892934675107632*T(5)+2038680/479233*T(6)*z^2+6681125407719/229664268289*T(6)*w-70022941096165/1377985609734*T(6)*z-131893934556420691/2009103018992172*T(6)-85280454901440/229664268289*T(7)*z^2-197828072606400/229664268289*T(7)*w-3*T(7)*z+155/9*T(7)+41665104/479233*w*z-130529907220320/229664268289*z^2-1012820277/1916932*w+45/4*z+115/2 norid[42]=-w^7+z^11-3*w^5*z^3-4*w^3*z^6+3*w*z^9-3*w^6*z-9*w^4*z^4+3*w^2*z^7-6*w^5*z^2+w^3*z^5-w^6 \end{verbatim} I can't get Macaulay2's integralClosure to work on this, so I pared it down to char 5 and still only got icFracP to work. I reduced the 7 fractions to get things of weights 9,3,6,5,7,0,2, of which of course only the 3 and 2 are necessary. \begin{verbatim} Macaulay 2, version 1.2 i1 : R=ZZ/5[w,z,MonomialOrder=>{Weights=>{11,7}}] i2 : I=ideal(w^7+3*w^6*z+w^6+3*w^5*z^3+6*w^5*z^2+9*w^4*z^4 +4*w^3*z^6-w^3*z^5-3*w^2*z^7-3*w*z^9-z^11) i3 : S=R/I i5 : F=icFracP(S) 5 3 4 4 2 2 5 5 3 3 6 4 2 4 7 3 2 w z - w z - 2w z + w z + w + w z + w*z + w z + w z + z + 2w z - 2o5 = {-------------------------------------------------------------------------- 4 2 4 7 3 2 4 6 w z - w z + z - 2w z + w + 2z + 2w --------------------------------------------------------------------------- 5 4 2 3 6 3 4 2 2 3 3 2 w*z + 2w - w z - 2z - 2w z - w*z + w z - w - w*z - w z --------------------------------------------------------------, 2 2 5 3 3 2 z - 2z + 2w - w*z + w z --------------------------------------------------------------------------- 2 2 5 3 3 6 3 4 2 2 3 w z + 2z - w + 2w*z z - w z + 2w*z + 2w z + w -----------------------, -----------------------------, 3 3 2 2 2 3 w + w*z + w z w z + w --------------------------------------------------------------------------- 5 4 6 3 2 2 5 3 3 w*z - w + z - 2w z - w z + z - w + w*z ---------------------------------------------, 4 2 2 5 3 3 w*z - w z + z + 2w + w*z --------------------------------------------------------------------------- 6 4 2 4 4 2 3 6 3 2 2 5 3 3 w*z - w z + 2w z + w - 2w z + z + 2w z - 2w z + 2z + 2w + 2w*z -----------------------------------------------------------------------, 1, 2 3 3 5 3 3 w z + w z + z - w + w*z --------------------------------------------------------------------------- 5 3 2 z - 2w - w z ---------------} 3 3 2 w + w*z + w z i7 : f0=F#0-w 5 3 3 6 2 4 7 3 2 5 4 2 3 3 - w + w z - 2w*z - 2w z - 2z + 2w z - 2w*z - 2w - w z - 2w z - 2w*o7 = --------------------------------------------------------------------------- 5 2 3 6 3 4 5 3 3 w*z + w z + 2z + w z + 2w*z + 2z + 2w + w*z --------------------------------------------------------------------------- 4 2 2 3 3 2 z - w z + 2w + w*z + w z ---------------------------- 2 - w z i8 : f1=F#1 2 2 5 3 3 w z + 2z - w + 2w*z o8 = ----------------------- 3 3 2 w + w*z + w z i9 : f2=F#2 6 3 4 2 2 3 z - w z + 2w*z + 2w z + w o9 = ----------------------------- 2 2 3 w z + w i11 : f3=F#3-z 4 2 3 3 4 2 2 5 3 3 - w + w z + w z - w*z - w z + z - w + w*z o11 = ------------------------------------------------ 4 2 2 5 3 3 w*z - w z + z + 2w + w*z i12 : f4=F#4 6 4 2 4 4 2 3 6 3 2 2 5 3 3 w*z - w z + 2w z + w - 2w z + z + 2w z - 2w z + 2z + 2w + 2w*z o12 = ----------------------------------------------------------------------- 2 3 3 5 3 3 w z + w z + z - w + w*z i13 : f5=F#5 o13 = 1 i14 : f6=F#6 5 3 2 z - 2w - w z o14 = --------------- 3 3 2 w + w*z + w z \end{verbatim} And normalP with the new ``withRing'' and ``noRed'' options gives \begin{verbatim} SINGULAR / A Computer Algebra System for Polynomial Computations / version 3-1-0 0< by: G.-M. Greuel, G. Pfister, H. Schoenemann \ Mar 2009 FB Mathematik der Universitaet, D-67653 Kaiserslautern \ > LIB "/home/leonada/presolve.lib"; > LIB "/home/leonada/normal.lib"; > ring r=5,(w,z),wp(11,7); > ideal i=w^7+3*w^6*z+w^6+3*w^5*z^3+6*w^5*z^2+9*w^4*z^4+4*w^3*z^6-w^3*z^5-3*w^2*z^7-3*w*z^9-z^11; > list norp=normalP(i,"withRing","noRed"); // 'normalP' created a list, say nor, of three lists: // nor[1] resp. nor[2] are lists of 1 ring(s) resp. ideal(s) // and nor[3] is a list of an intvec and an integer. // To see the result, type nor; // To access the i-th ring nor[1][i] give it a name, say Ri, and type e.g. def R1 = nor[1][1]; setring R1; norid; normap; // for the other rings type first setring r; (if r is the name of your // original basering) and then continue as for the first ring; // Ri/norid is the affine algebra of the normalization of the i-th // component r/P_i (where P_i is an associated prime of the input ideal) // and normap the normalization map from r to Ri/norid; // nor[2] is a list of 1 ideal(s), each ideal nor[2][i] consists of // elements g1..gk of r such that the gj/gk generate the integral // closure of r/P_i as (r/P_i)-module in the quotient field of r/P_i. // nor[3] shows the delta-invariant of each component and of the input // ideal (-1 means infinite, and 0 that r/P_i is normal). > norp; [1]: [1]: // characteristic : 5 // number of vars : 8 // block 1 : ordering dp // : names T(1) T(2) T(3) T(4) T(5) T(6) // block 2 : ordering wp // : names w z // : weights 11 7 // block 3 : ordering C [2]: [1]: _[1]=w2z6-w3z4-wz7+2w2z5+2w5+2w3z3-w4z _[2]=w4z3+2z9+w3z4+wz7-w4z2+2w2z5-2w5-w3z3+w4z _[3]=w6+2z9+w3z4+2w4z2-2w2z5+w5-w3z3-w4z _[4]=wz8-z9-w3z4+wz7+2w4z2-w2z5+2w5+w3z3-w4z _[5]=w3z5-z9+2w3z4+2wz7+2w4z2+w2z5+2w3z3-w4z _[6]=z10+2z9+wz7+w2z5+w5+2w3z3 _[7]=w5z+w3z4+2wz7-w5-w3z3+w4z [3]: [1]: 29 [2]: 29 > def R=norp[1][1]; > setring R; > normap; normap[1]=w normap[2]=z > norid; norid[1]=-2*T(1)*z-T(1)-T(2)*z+2*T(2)-T(3)*z+T(3)-T(4)-T(6)+w+2 norid[2]=T(1)*w+T(1)*z+2*T(1)+T(2)*z+T(2)-2*T(3)+2*T(4)-T(5)*z+2*T(6)+1 norid[3]=-T(1)*w+2*T(1)*z-T(1)-2*T(2)*w+T(2)*z+2*T(2)+T(3)*w+T(3)-T(4)*z-T(4)-T(6)*z-T(6)-2*w+2*z+2 norid[4]=-T(1)*w-T(2)*w+z^2 norid[5]=T(1)*z^2+T(1)*w-T(4)*w norid[6]=-T(1)*w-T(1)*z-2*T(1)+T(2)*z^2+2*T(2)*z-T(2)+2*T(3)-2*T(4)*z-2*T(4)-T(5)*w-2*T(6)*z-2*T(6)-z-1 norid[7]=T(1)*w-2*T(1)*z+T(1)-T(2)*z-2*T(2)-T(3)+T(4)*z^2+T(4)*z+T(4)-T(6)*w+T(6)*z+T(6)-2*z-2 norid[8]=T(1)^2-2*T(2)-T(3) norid[9]=T(1)*T(2)+T(1)+2*T(2)+T(3)-T(4) norid[10]=T(2)^2+2*T(2)+2*T(3)+T(4)-T(5)-1 norid[11]=T(1)*T(3)+2*T(1)-T(2)+T(3)+T(4)-T(5)-2 norid[12]=T(2)*T(3)-T(1)+2*T(3)+T(4)+2*T(5)-z norid[13]=T(3)^2-2*T(1)+T(2)-2*T(3)+2*T(4)+T(5)-T(6)-z+2 norid[14]=T(1)*T(4)-2*T(1)-2*T(3)+2*T(4)-T(5)-z+1 norid[15]=T(2)*T(4)+2*T(1)-2*T(2)-2*T(3)-2*T(4)+T(5)-T(6)+z+2 norid[16]=T(3)*T(4)-T(1)*z+2*T(1)+2*T(2)+T(3)+T(5)+T(6)+z+1 norid[17]=T(4)^2-2*T(1)*z+2*T(1)-T(2)*z+T(2)-2*T(3)-2*T(4)+2*T(6)+2*z+1 norid[18]=T(1)*T(5)+2*T(1)-T(2)-2*T(3)-2*T(4)-2*T(5)-T(6)+z+1 norid[19]=T(2)*T(5)-T(1)*z+2*T(1)-T(2)-2*T(3)+T(4)+T(5)+T(6)-z-1 norid[20]=T(3)*T(5)+T(1)*z-2*T(1)-T(2)*z-2*T(4)-2*T(6) norid[21]=T(4)*T(5)+2*T(1)*z-2*T(1)-T(2)*z-T(2)-2*T(3)+T(4)+T(5)-T(6)-w+z-2 norid[22]=T(5)^2+2*T(1)*z-T(1)-2*T(2)*z-T(4)*z-T(4)+T(5)+T(6)+w-2*z-2 norid[23]=T(1)*T(6)-2*T(1)*z+T(1)-T(2)*z+T(2)-T(3)-T(4)+2*T(6)-2*z-2 norid[24]=T(2)*T(6)-T(2)*z-T(2)-2*T(3)-T(5)-w-z-2 norid[25]=T(3)*T(6)+T(1)-2*T(2)*z+2*T(2)-T(3)-T(4)*z+2*T(4)+2*T(6)+w norid[26]=T(4)*T(6)-T(1)*w+T(1)*z+T(1)+T(2)*z+2*T(2)+T(3)-2*T(4)*z-T(4)-T(5)+2*T(6)+2*z-2 norid[27]=T(5)*T(6)-T(1)*w-T(1)-2*T(2)*z+T(2)-2*T(3)+2*T(4)*z-2*T(4)+T(5)-T(6)-z^2+2*w+z norid[28]=T(6)^2-T(1)*z^2-T(1)*w+2*T(1)*z+2*T(1)-2*T(2)*z+T(2)-2*T(3)*w+2*T(3)+2*T(4)*z+2*T(4)-T(5)-T(6)-z^2-2*w+2*z-1 norid[29]=w^7-z^11-2*w^5*z^3-w^3*z^6+2*w*z^9-2*w^6*z-w^4*z^4+2*w^2*z^7+w^5*z^2-w^3*z^5+w^6 > option(redSB); > ideal j=std(norid);j; j[1]=w^7-z^11-2*w^5*z^3-w^3*z^6+2*w*z^9-2*w^6*z-w^4*z^4+2*w^2*z^7+w^5*z^2-w^3*z^5+w^6 j[2]=T(6)*w^2*z^2+2*T(6)*w^3+T(6)*w*z^3+T(6)*w^2*z-w^4-2*w^2*z^3+w^3*z+2*w*z^4+w^2*z^2-2*z^5+w^3-2*w*z^3 j[3]=T(6)*w*z^5-T(6)*z^6+T(6)*w^3*z-2*T(6)*w*z^4-T(6)*w^3-2*T(6)*w*z^3-2*T(6)*w^2*z-w^3*z^3-2*w*z^6+2*w^4*z+2*w^2*z^4+z^7+2*w*z^5-w^4+2*w^3*z+2*w^2*z^2-z^5-2*w^3-w*z^3 j[4]=T(6)*z^7-T(6)*w^4-T(6)*z^6-T(6)*w^3*z-T(6)*w*z^4+2*T(6)*w^3+2*T(6)*w*z^3+2*T(6)*w^2*z-w^2*z^5-2*z^8+2*w^3*z^3+w*z^6-w^2*z^4+2*z^7+w^3*z^2-2*w*z^5-w^4+w^2*z^3-2*z^6+2*w^3*z+w*z^4+w^2*z^2+z^5+2*w^3+w*z^3 j[5]=T(6)*w^5-2*T(6)*w^4*z-2*T(6)*z^6-2*T(6)*w^3*z+2*T(6)*w*z^3+2*T(6)*w^2*z-z^9+w^5*z-w^3*z^4-w*z^7-w^4*z^2-w^2*z^5-z^8-w^5+w^3*z^3+w*z^6+2*w^2*z^4+z^7+2*w^3*z^2+w^2*z^3+2*z^6-w^3*z-w*z^4+z^5+2*w^3+w*z^3 j[6]=T(5)*z^3+T(5)*w*z-2*T(6)*w*z^3+T(6)*w^2*z+2*T(6)*z^4+2*T(6)*w*z^2+2*T(6)*w^2+2*T(6)*z^3+2*T(6)*w*z+2*w^3*z-w*z^4-w^2*z^2-2*z^5-w^3-w*z^3-w^2*z+2*z^4+w*z^2+w^2+z^3+w*z j[7]=T(5)*w*z^2+T(5)*w^2-T(6)*w*z^3-2*T(6)*w^2*z+T(6)*w*z^2+T(6)*w^2+w^3*z+2*w*z^4-2*w^2*z^2+z^5+w*z^3+2*z^4-2*w*z^2+w^2 j[8]=T(5)*w^2*z+2*T(6)*w*z^4-2*T(6)*z^5+2*T(6)*w^3+2*T(6)*w*z^3-T(6)*z^4+2*T(6)*w*z^2-2*T(6)*w^2-2*w^3*z^2+w*z^5-w^4-w^2*z^3+2*z^6-w^3*z+2*w*z^4+2*w^2*z^2-2*w^3-2*w^2*z-z^4-w*z^2 j[9]=T(5)*w^3+T(6)*z^5+2*T(6)*w*z^3+T(6)*w^2*z-w^2*z^3-2*z^6+w*z^4+2*w^2*z^2-2*z^5-2*w*z^3 j[10]=T(4)*w-T(5)*w^2+2*T(5)*w*z+2*T(5)*z^2+2*T(5)*w-2*T(6)*w*z^3+T(6)*w^2*z+2*T(6)*z^4-T(6)*w*z^2+T(6)*w^2-T(6)*z^3+2*T(6)*w*z+T(6)*z^2+T(6)*w+2*w^3*z-w*z^4-w^2*z^2-2*z^5+2*w^3+w^2*z+2*z^4-2*w*z^2-2*w^2-2*w*z-2*z^2-2*w j[11]=T(4)*z^2-2*T(5)*z^2-2*T(5)*w-T(6)*w*z^2-2*T(6)*w^2+T(6)*z^3-2*T(6)*w*z-T(6)*z^2-T(6)*w+w^3+2*w*z^3+2*w^2*z-z^4-2*w*z^2-2*w^2-2*z^3+2*w*z-2*z^2-2*w j[12]=T(3)*w+T(5)*w^2-2*T(5)*w*z-T(5)*z^2+2*T(6)*w*z^3-T(6)*w^2*z-2*T(6)*z^4-T(6)*w*z^2-2*T(6)*z^3-T(6)*w*z+T(6)*z^2+2*T(6)*w-2*w^3*z+w*z^4+w^2*z^2+2*z^5-w*z^3-2*w^2*z+z^4-2*w*z^2-w^2-2*z^3-w*z+2*z^2+w j[13]=T(3)*z^2+T(5)*z^2-2*T(6)*w*z^2+T(6)*w^2+2*T(6)*z^3+T(6)*w*z-T(6)*w+2*w^3-w*z^3-w^2*z-2*z^4+w*z^2-w^2+2*z^3+2*w*z-z^2 j[14]=T(2)*z+T(4)*z-T(5)*w^2+2*T(5)*w*z-T(5)*w-2*T(5)*z-2*T(6)*w*z^3+T(6)*w^2*z+2*T(6)*z^4-2*T(6)*w*z^2-T(6)*w^2+T(6)*z^2-2*T(6)*w+T(6)*z+2*w^3*z-w*z^4-w^2*z^2-2*z^5-2*w^3+2*w*z^3-2*w^2*z+z^4+w*z^2+z^3+2*w*z+w-2*z j[15]=T(2)*w-2*T(5)*w^2-T(5)*w*z+T(5)*z^2-T(5)*w+T(6)*w*z^3+2*T(6)*w^2*z-T(6)*z^4-T(6)*w*z^2-T(6)*w^2+2*T(6)*z^3+T(6)*w*z+T(6)*w-w^3*z-2*w*z^4-2*w^2*z^2+z^5-2*w^3-2*w*z^3-2*w*z^2+w^2-2*z^3-2*w*z-2*w j[16]=T(1)-2*T(2)-2*T(3)*z-T(3)-2*T(4)*z+T(4)-T(5)*w^2+2*T(5)*w*z-T(5)*z^2-2*T(5)*w-2*T(6)*w*z^3+T(6)*w^2*z+2*T(6)*z^4-2*T(6)*w^2-2*T(6)*z^3-T(6)*w*z-2*T(6)*z^2-2*T(6)*w-2*T(6)*z+T(6)+2*w^3*z-w*z^4-w^2*z^2-2*z^5+w^3-2*w*z^3-w^2*z-2*z^4-w^2-2*w*z-z^2+2*w-z-2 j[17]=T(6)^2+T(3)*z-T(3)+2*T(4)*z-T(5)*w^2+2*T(5)*w*z+T(5)*z^2+T(5)*w-T(5)*z-T(5)-2*T(6)*w*z^3+T(6)*w^2*z+2*T(6)*z^4+T(6)*w*z^2+2*T(6)*z^3+T(6)*w*z-2*T(6)*z^2+T(6)*w+2*T(6)+2*w^3*z-w*z^4-w^2*z^2-2*z^5+w*z^3+2*w^2*z-z^4+2*w*z^2+2*w^2-z^3-w*z+2*z^2+2*z-2 j[18]=T(5)*T(6)-T(2)-2*T(3)*z+2*T(3)+2*T(4)*z-T(4)-T(5)*w^2+2*T(5)*w*z-2*T(5)*z^2+2*T(5)*w+T(5)*z+T(5)-2*T(6)*w*z^3+T(6)*w^2*z+2*T(6)*z^4+2*T(6)*w*z^2+2*T(6)*w^2+T(6)*z^3-2*T(6)*w*z-2*T(6)*w+2*w^3*z-w*z^4-w^2*z^2-2*z^5-w^3-w*z^3-w*z^2-2*w^2-z^3-w*z+2*z^2-2*w+z-2 j[19]=T(4)*T(6)-T(2)-2*T(3)*z+2*T(3)+T(4)*z-2*T(4)-2*T(5)*w^2-T(5)*w*z+2*T(5)*z^2+T(5)*z-T(5)+T(6)*w*z^3+2*T(6)*w^2*z-T(6)*z^4+2*T(6)*w*z^2-T(6)*z^3+2*T(6)*w*z-2*T(6)*z^2+T(6)*w-2*T(6)*z+T(6)-w^3*z-2*w*z^4-2*w^2*z^2+z^5+2*w*z^3-w^2*z-2*z^4-w*z^2+2*w^2-z^3+2*w*z+2*z^2+w+z j[20]=T(3)*T(6)-T(2)+2*T(3)*z-2*T(4)*z+T(4)-T(5)*w^2+2*T(5)*w*z+T(5)*z^2+T(5)*z-2*T(6)*w*z^3+T(6)*w^2*z+2*T(6)*z^4+T(6)*w*z^2+2*T(6)*z^3+T(6)*w*z-T(6)*z^2-2*T(6)*w-T(6)*z+T(6)+2*w^3*z-w*z^4-w^2*z^2-2*z^5+w*z^3+2*w^2*z-z^4+2*w*z^2+w^2+2*z^3+w*z+z^2+w+2*z+2 j[21]=T(2)*T(6)-T(2)-2*T(3)+T(4)*z-T(5)*w^2+2*T(5)*w*z-T(5)*w-2*T(5)*z-T(5)-2*T(6)*w*z^3+T(6)*w^2*z+2*T(6)*z^4-2*T(6)*w*z^2-T(6)*w^2+T(6)*z^2-2*T(6)*w+T(6)*z+2*w^3*z-w*z^4-w^2*z^2-2*z^5-2*w^3+2*w*z^3-2*w^2*z+z^4+w*z^2+z^3+2*w*z+2*z-2 j[22]=T(5)^2-2*T(2)-T(3)-2*T(4)*z-2*T(5)*z^2-2*T(5)*w-T(5)*z+T(5)-T(6)*w*z^2-2*T(6)*w^2+T(6)*z^3-2*T(6)*w*z-T(6)*z^2-T(6)*z+2*T(6)+w^3+2*w*z^3+2*w^2*z-z^4-2*w*z^2-2*w^2-2*z^3+2*w*z-2*z^2-w+1 j[23]=T(4)*T(5)-2*T(3)*z+T(3)+T(4)*z-2*T(4)+2*T(5)*z^2+2*T(5)*w+T(5)*z+T(5)+T(6)*w*z^2+2*T(6)*w^2-T(6)*z^3+2*T(6)*w*z+T(6)*z^2+T(6)*z+T(6)-w^3-2*w*z^3-2*w^2*z+z^4+2*w*z^2+2*w^2+2*z^3-2*w*z+2*z^2-2*w-z-1 j[24]=T(3)*T(5)+T(2)+2*T(3)*z-2*T(3)-T(4)*z+T(5)*w^2-2*T(5)*w*z+T(5)*w+2*T(5)*z+2*T(6)*w*z^3-T(6)*w^2*z-2*T(6)*z^4+2*T(6)*w*z^2+T(6)*w^2-T(6)*z^2+2*T(6)*w-T(6)*z-2*w^3*z+w*z^4+w^2*z^2+2*z^5+2*w^3-2*w*z^3+2*w^2*z-z^4-w*z^2-z^3-2*w*z+2*w+2*z+1 j[25]=T(2)*T(5)-2*T(2)-2*T(3)*z+2*T(4)*z-T(4)-2*T(5)*w^2-T(5)*w*z-2*T(5)*w+T(5)*z+T(5)+T(6)*w*z^3+2*T(6)*w^2*z-T(6)*z^4+T(6)*w*z^2-2*T(6)*w^2+2*T(6)*z^2+T(6)*w+2*T(6)*z-T(6)-w^3*z-2*w*z^4-2*w^2*z^2+z^5+w^3-w*z^3+w^2*z+2*z^4+2*w*z^2+2*z^3-w*z-w-2 j[26]=T(4)^2+2*T(3)*z+T(4)*z+T(4)-2*T(5)*w^2-T(5)*w*z-2*T(5)*z^2+T(5)*w+T(6)*w*z^3+2*T(6)*w^2*z-T(6)*z^4+T(6)*w^2+T(6)*z^3-2*T(6)*w*z+T(6)*z^2+T(6)*w+T(6)*z-w^3*z-2*w*z^4-2*w^2*z^2+z^5+2*w^3+w*z^3-2*w^2*z+z^4-2*w^2+w*z-2*z^2-2*w j[27]=T(3)*T(4)+T(2)-2*T(3)*z-2*T(3)+2*T(4)*z-2*T(4)-2*T(5)*w^2-T(5)*w*z-2*T(5)*w+T(5)*z+T(5)+T(6)*w*z^3+2*T(6)*w^2*z-T(6)*z^4+T(6)*w*z^2-2*T(6)*w^2+2*T(6)*z^2+T(6)*w+2*T(6)*z-T(6)-w^3*z-2*w*z^4-2*w^2*z^2+z^5+w^3-w*z^3+w^2*z+2*z^4+2*w*z^2+2*z^3-w*z-w+2*z j[28]=T(2)*T(4)+2*T(2)-T(3)*z-T(4)*z+T(4)+2*T(5)*w^2+T(5)*w*z+2*T(5)*z^2-T(5)*w+T(5)-T(6)*w*z^3-2*T(6)*w^2*z+T(6)*z^4-T(6)*w^2-T(6)*z^3+2*T(6)*w*z-T(6)*z^2-T(6)*w-T(6)*z+2*T(6)+w^3*z+2*w*z^4+2*w^2*z^2-z^5-2*w^3-w*z^3+2*w^2*z-z^4+2*w^2-w*z+2*z^2+w-2*z+1 j[29]=T(3)^2+2*T(2)+T(3)*z+T(3)+T(4)*z-T(4)-2*T(5)*w^2-T(5)*w*z-2*T(5)*z^2+T(5)*w+T(5)+T(6)*w*z^3+2*T(6)*w^2*z-T(6)*z^4+T(6)*w^2+T(6)*z^3-2*T(6)*w*z+T(6)*z^2+T(6)*w+T(6)*z+T(6)-w^3*z-2*w*z^4-2*w^2*z^2+z^5+2*w^3+w*z^3-2*w^2*z+z^4-2*w^2+w*z-2*z^2-w+2*z-2 j[30]=T(2)*T(3)-2*T(2)-2*T(3)*z+T(3)-2*T(4)*z+2*T(4)-T(5)*w^2+2*T(5)*w*z-T(5)*z^2-2*T(5)*w+2*T(5)-2*T(6)*w*z^3+T(6)*w^2*z+2*T(6)*z^4-2*T(6)*w^2-2*T(6)*z^3-T(6)*w*z-2*T(6)*z^2-2*T(6)*w-2*T(6)*z+T(6)+2*w^3*z-w*z^4-w^2*z^2-2*z^5+w^3-2*w*z^3-w^2*z-2*z^4-w^2-2*w*z-z^2+2*w-2*z-2 j[31]=T(2)^2+2*T(2)+2*T(3)+T(4)-T(5)-1 \end{verbatim} again having no clue that this is a disguised elliptic curve example, whereas without the ``noRed'' option, the same set of fractions gives rise to a supposedly minimal presentation which is still not minimal enough: \begin{verbatim} list norp=normalP(i,"withRing"); > norp; [1]: [1]: // characteristic : 5 // number of vars : 4 // block 1 : ordering dp // : names T(2) T(5) T(6) // block 2 : ordering wp // : names z // : weights 7 // block 3 : ordering C [2]: [1]: _[1]=w2z6-w3z4-wz7+2w2z5+2w5+2w3z3-w4z _[2]=w4z3+2z9+w3z4+wz7-w4z2+2w2z5-2w5-w3z3+w4z _[3]=w6+2z9+w3z4+2w4z2-2w2z5+w5-w3z3-w4z _[4]=wz8-z9-w3z4+wz7+2w4z2-w2z5+2w5+w3z3-w4z _[5]=w3z5-z9+2w3z4+2wz7+2w4z2+w2z5+2w3z3-w4z _[6]=z10+2z9+wz7+w2z5+w5+2w3z3 _[7]=w5z+w3z4+2wz7-w5-w3z3+w4z [3]: [1]: 29 [2]: 29 > def R=norp[1][1]; > setring R; > normap; normap[1]=-T(2)^3-2*T(2)^2+T(2)*T(5)+T(2)*T(6)-T(2)*z-2*T(2)-T(5)-T(6)-2*z normap[2]=z > norid; norid[1]=T(2)^4*z-2*T(2)^4-2*T(2)^3-T(2)^2*T(5)*z+2*T(2)^2*T(5)+2*T(2)*T(5)*z-2*T(2)*T(5)+T(2)*T(6)*z+2*T(2)*T(6)+T(2)*z^2+2*T(2)*z+T(2)+T(5)*z-2*T(6)*z-2*T(6)+z^2-2*z-2 norid[2]=2*T(2)^7-2*T(2)^6+T(2)^5*T(5)-2*T(2)^5*T(6)+2*T(2)^5*z-T(2)^4*T(5)+2*T(2)^3*T(5)^2+2*T(2)^3*T(5)*T(6)-2*T(2)^4*z-2*T(2)^4-2*T(2)^3*T(5)*z-T(2)^3*T(5)-2*T(2)^2*T(5)^2+T(2)^2*T(5)*T(6)-2*T(2)^2*T(6)^2+2*T(2)^3*z+T(2)^2*T(5)*z-2*T(2)^2*T(5)-2*T(2)*T(5)^2-T(2)^2*T(6)+2*T(2)*T(6)^2+2*T(2)^2*z^2+2*T(2)^2*z-2*T(2)*T(5)*z+T(2)*T(5)+2*T(5)^2-2*T(2)*T(6)*z+2*T(5)*T(6)-2*T(2)*z^2+T(2)*z+T(2)+2*T(5)*z+2*T(5)+T(6)*z-2*T(6)+z^2-z+1 norid[3]=2*T(2)^6-2*T(2)^5+T(2)^4*T(5)-2*T(2)^4*T(6)-T(2)^4*z-2*T(2)^4-T(2)^3*T(5)+2*T(2)^2*T(5)^2+2*T(2)^2*T(5)*T(6)-T(2)^3*z+T(2)^2*T(5)*z-T(2)^2*T(5)-2*T(2)*T(5)^2-T(2)^2*T(6)+T(2)*T(5)*T(6)-2*T(2)*T(6)^2+2*T(2)^2-T(2)*T(5)*z-T(2)*T(5)+2*T(2)*T(6)*z-T(2)*T(6)+2*T(5)*T(6)+2*T(6)^2-T(2)*z^2-2*T(2)+2*T(5)*z+2*T(5)-T(6)+2*z+2 norid[4]=2*T(2)^4*z-2*T(2)^4-T(2)^3*z-T(2)^3-2*T(2)^2*T(5)*z+2*T(2)^2*T(5)+T(2)^2*T(6)+2*T(2)^2*z+2*T(2)^2+T(2)*T(5)+2*T(2)*T(6)*z-2*T(2)*T(6)+2*T(2)*z^2-T(2)*z+2*T(2)-2*T(5)*z+2*T(5)+T(6)-z-1 norid[5]=T(2)^6-2*T(2)^4*T(5)-T(2)^4*T(6)+2*T(2)^4*z^2-T(2)^4*z+2*T(2)^4+T(2)^2*T(5)^2-T(2)^3*T(6)+T(2)^2*T(5)*T(6)-T(2)^3*z^2+2*T(2)^3*z-2*T(2)^3-2*T(2)^2*T(5)*z^2+T(2)^2*T(5)*z+2*T(2)^2*T(6)-T(2)*T(5)*T(6)-T(2)*T(6)^2-2*T(2)^2*z^2-2*T(2)^2*z+T(2)^2+T(2)*T(5)*z+2*T(2)*T(5)-T(5)^2+2*T(2)*T(6)*z^2-2*T(2)*T(6)*z-T(2)*T(6)+T(6)^2+2*T(2)*z^3+2*T(2)*z^2-T(2)*z+2*T(2)+2*T(5)*z^2+T(5)*z-2*T(6)*z+T(6)+z^3+1 norid[6]=T(2)^3*T(5)+2*T(2)^3*z+2*T(2)^2*T(5)-T(2)*T(5)^2-T(2)*T(5)*T(6)+T(2)^2*z-T(2)*T(5)*z+2*T(2)*T(5)+T(5)^2+T(5)*T(6)+T(2)*z^2-T(2)*z-T(5)*z+2*z^2-2*z norid[7]=T(2)^4*z+2*T(2)^4+T(2)^3*T(6)-T(2)^3*z^2+T(2)^3*z-2*T(2)^3-T(2)^2*T(5)*z-2*T(2)^2*T(5)+2*T(2)^2*T(6)-T(2)*T(5)*T(6)-T(2)*T(6)^2+2*T(2)^2*z^2+T(2)^2*z+2*T(2)^2+T(2)*T(5)*z^2+T(2)*T(5)*z+T(2)*T(5)+2*T(2)*T(6)*z-T(2)*T(6)+T(5)*T(6)+T(6)^2-2*T(2)*z^2+T(2)*z+T(2)+T(5)*z^2-2*T(5)*z+T(5)-T(6)*z^2+2*T(6)*z-2*T(6)-z^3-2*z+2 norid[8]=T(2)^8-T(2)^7-2*T(2)^6*T(5)+2*T(2)^6+T(2)^5*T(5)+T(2)^4*T(5)^2+2*T(2)^5*T(6)+2*T(2)^5*z-T(2)^5-T(2)^4*T(5)-T(2)^4*T(6)-2*T(2)^3*T(5)*T(6)+T(2)^4-2*T(2)^3*T(5)*z+T(2)^3*T(5)-2*T(2)^2*T(5)^2-2*T(2)^3*T(6)+T(2)^2*T(6)^2-2*T(2)^3-T(2)^2*T(5)*z-T(2)^2*T(5)+2*T(2)^2*T(6)*z-2*T(2)^2*T(6)+2*T(2)*T(5)*T(6)+T(2)^2*z^2+2*T(2)^2*z-2*T(2)^2+2*T(2)*T(5)*z+T(5)^2+T(2)*T(6)*z-T(2)*T(6)+T(2)*z^2-2*T(2)*z+T(2)+T(5)*z-T(5)-2*T(6)-z^2-2 norid[9]=2*T(2)^5+T(2)^4-2*T(2)^3*T(5)-2*T(2)^3-2*T(2)^2*T(5)+2*T(2)^2*T(6)+2*T(2)^2*z+2*T(2)^2+T(2)*T(5)+2*T(2)*T(6)-2*T(2)*z-T(2)-2*T(5)+T(6)+2*z-2 norid[10]=T(2)^7-T(2)^6-2*T(2)^5*T(5)-T(2)^5+T(2)^4*T(5)+T(2)^3*T(5)^2+2*T(2)^4*T(6)+2*T(2)^4*z+T(2)^4+T(2)^3*T(5)-T(2)^3*T(6)-2*T(2)^2*T(5)*T(6)+2*T(2)^3*z+T(2)^3-2*T(2)^2*T(5)*z+T(2)^2*T(5)-T(2)*T(5)^2+T(2)*T(6)^2+T(2)^2*z-2*T(2)^2+2*T(2)*T(5)*z+T(2)*T(5)+2*T(2)*T(6)*z+T(2)*T(6)+T(5)*T(6)+T(2)*z^2-T(2)*z+T(2)+T(5)*z+2*T(5)-2*T(6)*z-2*z^2+z-1 norid[11]=T(2)^6-T(2)^5-2*T(2)^4*T(5)+2*T(2)^4+T(2)^3*T(5)+T(2)^2*T(5)^2+2*T(2)^3*T(6)+2*T(2)^3*z+2*T(2)^2*T(5)-T(2)^2*T(6)-2*T(2)*T(5)*T(6)-T(2)^2*z-2*T(2)*T(5)*z+T(2)*T(5)-2*T(2)*T(6)+T(6)^2-2*T(2)*z+T(2)-2*T(5)+2*T(6)*z+z^2-2*z+1 norid[12]=2*T(2)^7+T(2)^5*T(5)+T(2)^5-2*T(2)^4*T(5)+2*T(2)^3*T(5)^2-T(2)^4*T(6)-T(2)^4*z+2*T(2)^4-T(2)^3*T(5)+2*T(2)^2*T(5)^2+T(2)^2*T(5)*T(6)+T(2)^3*z+T(2)^2*T(5)*z+2*T(2)^2*T(5)-2*T(2)*T(5)^2-T(2)^2*T(6)-2*T(2)*T(5)*T(6)+2*T(2)*T(6)^2+2*T(2)^2*z+2*T(2)*T(5)*z+2*T(2)*T(5)-2*T(5)^2-T(2)*T(6)*z+T(2)*T(6)+2*T(5)*T(6)+2*T(2)*z^2-T(2)*z+2*T(2)+T(5)*z+T(6)*z+T(6)+z^2-z norid[13]=2*T(2)^6+T(2)^4*T(5)+2*T(2)^4*z+T(2)^4-2*T(2)^3*T(5)+2*T(2)^2*T(5)^2-T(2)^3*T(6)-2*T(2)^3*z-2*T(2)^3-2*T(2)^2*T(5)*z+2*T(2)^2*T(5)+2*T(2)*T(5)^2+T(2)*T(5)*T(6)-2*T(2)^2*z+2*T(2)^2+T(2)*T(5)*z+2*T(2)*T(6)*z-T(2)*T(6)-2*T(5)*T(6)+2*T(6)^2+2*T(2)*z^2+2*T(2)*z+T(2)+T(5)-T(6)*z-T(6)-2*z^2+z-1 norid[14]=T(2)^6+T(2)^5-2*T(2)^4*T(5)-T(2)^4*z+T(2)^4+2*T(2)^3*T(5)+T(2)^2*T(5)^2+2*T(2)^3*T(6)+T(2)^2*T(5)*z+2*T(2)^2*T(5)+2*T(2)*T(5)^2+T(2)^2*T(6)-2*T(2)*T(5)*T(6)+2*T(2)^2*z-T(2)^2-2*T(2)*T(5)*z-T(2)*T(5)+T(5)^2-T(2)*T(6)*z+2*T(2)*T(6)-2*T(5)*T(6)+T(6)^2-T(2)*z^2+2*T(2)*z+2*T(2)+2*T(5)*z+2*T(6)*z+2*T(6)-2*z^2-2*z-2 norid[15]=2*T(2)^4*T(5)-T(2)^4-T(2)^3*T(5)-2*T(2)^2*T(5)^2+2*T(2)^3-T(2)^2*T(5)+2*T(2)*T(5)*T(6)+2*T(2)^2+2*T(2)*T(5)*z-T(2)*T(5)+2*T(5)^2-T(2)*T(6)-T(2)*z+2*T(2)+T(5)*z+2*T(5)+1 norid[16]=2*T(2)^4*z+T(2)^4-T(2)^3*z-2*T(2)^2*T(5)*z-T(2)^2*T(5)-2*T(2)^2*z-T(2)^2-2*T(2)*T(5)+2*T(2)*T(6)*z+T(2)*T(6)+2*T(2)*z^2-T(2)*z-T(2)+2*T(5)*z-2*T(5)-T(6)+z^2-z+1 norid[17]=2*T(2)^4*z+T(2)^4+2*T(2)^3*T(5)-T(2)^3*z-2*T(2)^2*T(5)*z-2*T(2)^2*T(5)-2*T(2)*T(5)^2-2*T(2)^2*z-2*T(2)^2-T(2)*T(5)+2*T(2)*T(6)*z+T(2)*T(6)+2*T(5)*T(6)+2*T(2)*z^2-2*T(2)-T(5)*z-T(5)+z^2-2 norid[18]=T(2)^4*z-T(2)^4-T(2)^3*T(5)+2*T(2)^3*z+2*T(2)^3-T(2)^2*T(5)*z-2*T(2)^2*T(5)+T(2)*T(5)^2-T(2)^2*z-2*T(2)^2-2*T(2)*T(5)+T(5)^2+T(2)*T(6)*z-2*T(2)*T(6)-T(5)*T(6)+T(2)*z^2-2*T(2)*z-2*T(2)-2*T(6)-2*z^2+1 norid[19]=2*T(2)^4*z-2*T(2)^3*z+2*T(2)^3-2*T(2)^2*T(5)*z+2*T(2)^2*z-T(2)^2+T(2)*T(5)*z-2*T(2)*T(5)+T(5)^2+2*T(2)*T(6)*z+2*T(2)*z^2-2*T(2)*z+T(5)*z+T(5)-T(6)*z-T(6)+2 norid[20]=2*T(2)^4*T(6)+T(2)^4*z+2*T(2)^4-T(2)^3*T(6)-2*T(2)^2*T(5)*T(6)+2*T(2)^3*z+T(2)^3-T(2)^2*T(5)*z-2*T(2)^2*T(5)-2*T(2)^2*T(6)+2*T(2)*T(6)^2-T(2)^2*z+T(2)^2-2*T(2)*T(5)-2*T(2)*T(6)*z+2*T(5)*T(6)+T(2)*z^2+2*T(2)*z+2*T(2)+T(5)*z-2*T(5)+T(6)*z-T(6)-2*z^2+2*z-2 norid[21]=T(2)^4*z+T(2)^4-2*T(2)^3*T(6)+T(2)^3*z-2*T(2)^3-T(2)^2*T(5)*z-T(2)^2*T(5)+T(2)^2*T(6)+2*T(2)*T(5)*T(6)-2*T(2)^2*z-2*T(2)^2+T(2)*T(5)*z-T(2)*T(5)+T(2)*T(6)*z-T(2)*T(6)-2*T(6)^2+T(2)*z^2-T(2)-T(5)+2*T(6)*z-2*T(6)+2*z^2+2*z+1 norid[22]=2*T(2)^7-2*T(2)^6+T(2)^5*T(5)-2*T(2)^5*T(6)+2*T(2)^5*z-T(2)^4*T(5)+2*T(2)^3*T(5)^2+2*T(2)^3*T(5)*T(6)-2*T(2)^4-2*T(2)^3*T(5)*z-T(2)^3*T(5)-2*T(2)^2*T(5)^2+T(2)^3*T(6)+T(2)^2*T(5)*T(6)-2*T(2)^2*T(6)^2-2*T(2)^3*z-T(2)^2*T(5)*z-2*T(2)^2*T(5)-2*T(2)*T(5)^2+2*T(2)^2*T(6)-T(2)*T(5)*T(6)+2*T(2)*T(6)^2+2*T(2)^2*z^2+2*T(2)^2*z-2*T(2)^2+T(2)*T(5)*z+T(2)*T(5)+2*T(5)^2+T(5)*T(6)+T(6)^2+T(2)*z+T(2)+2*T(5)*z-2*T(5)-T(6)*z-T(6)-z^2+2*z-1 norid[23]=2*T(2)^7-2*T(2)^6+T(2)^5*T(5)-2*T(2)^5*T(6)+2*T(2)^5*z-T(2)^4*T(5)+2*T(2)^3*T(5)^2+2*T(2)^3*T(5)*T(6)+2*T(2)^4*z+T(2)^4-2*T(2)^3*T(5)*z-T(2)^3*T(5)-2*T(2)^2*T(5)^2+T(2)^2*T(5)*T(6)-2*T(2)^2*T(6)^2-2*T(2)^3*z-T(2)^3+2*T(2)^2*T(5)*z-2*T(2)*T(5)^2-T(2)^2*T(6)+2*T(2)*T(6)^2+2*T(2)^2*z^2+T(2)^2*z+T(2)^2-2*T(2)*T(5)+2*T(5)^2+2*T(2)*T(6)*z-T(2)*T(6)+T(5)*T(6)+2*T(2)*z^2-T(2)*z-T(6)*z+2*T(6)+2*z^2-z+1 norid[24]=2*T(2)^7+2*T(2)^6+T(2)^5*T(5)-2*T(2)^5*T(6)+2*T(2)^5*z+T(2)^5+T(2)^4*T(5)+2*T(2)^3*T(5)^2+T(2)^4*T(6)+2*T(2)^3*T(5)*T(6)-2*T(2)^4*z^2-2*T(2)^3*T(5)*z+2*T(2)^3*T(5)+2*T(2)^2*T(5)^2-2*T(2)^2*T(6)^2+T(2)^3*z^2+T(2)^3*z+2*T(2)^2*T(5)*z^2-T(2)^2*T(5)*z-T(2)*T(5)^2+T(2)^2*T(6)+2*T(2)*T(5)*T(6)-2*T(2)*T(6)^2-T(2)^2*z^2+2*T(2)^2-2*T(2)*T(5)*z+2*T(2)*T(5)+2*T(5)^2-2*T(2)*T(6)*z^2+T(2)*T(6)*z-T(2)*T(6)+T(5)*T(6)-2*T(6)^2-2*T(2)*z^3+2*T(2)*z^2-2*T(5)*z^2-2*T(5)*z+2*T(5)+T(6)*z+2*T(6)-z^3-2*z^2+2*z+1 norid[25]=T(2)^21-T(2)^20-2*T(2)^19*T(5)-2*T(2)^19*T(6)+2*T(2)^19*z-2*T(2)^19-2*T(2)^18*T(5)+T(2)^17*T(5)^2-2*T(2)^18*T(6)+2*T(2)^17*T(5)*T(6)+T(2)^17*T(6)^2+2*T(2)^18-2*T(2)^17*T(5)*z-2*T(2)^16*T(5)^2-2*T(2)^17*T(6)*z+T(2)^16*T(5)*T(6)-2*T(2)^16*T(6)^2+T(2)^17*z^2+T(2)^17*z+2*T(2)^17+T(2)^16*T(5)*z+T(2)^15*T(5)^2+T(2)^16*T(6)*z+2*T(2)^15*T(5)*T(6)+T(2)^15*T(6)^2+T(2)^16*z^2+T(2)^16*z+T(2)^15*T(5)*z-T(2)^15*T(5)-T(2)^11*T(5)^5+T(2)^15*T(6)*z-T(2)^15*T(6)-T(2)^11*T(6)^5-2*T(2)^15*z^3-T(2)^15*z^2-2*T(2)^15*z-2*T(2)^15+T(2)^14*T(5)+T(2)^10*T(5)^5+2*T(2)^9*T(5)^6+T(2)^14*T(6)+2*T(2)^9*T(5)^5*T(6)+T(2)^10*T(6)^5+2*T(2)^9*T(5)*T(6)^5+2*T(2)^9*T(6)^6-T(2)^14*z+T(2)^14+T(2)^13*T(5)+2*T(2)^12*T(5)^2-2*T(2)^9*T(5)^5*z+2*T(2)^9*T(5)^5+2*T(2)^8*T(5)^6-T(2)^7*T(5)^7+T(2)^13*T(6)-T(2)^12*T(5)*T(6)+2*T(2)^8*T(5)^5*T(6)-2*T(2)^7*T(5)^6*T(6)+2*T(2)^12*T(6)^2-T(2)^7*T(5)^5*T(6)^2-2*T(2)^9*T(6)^5*z+2*T(2)^9*T(6)^5+2*T(2)^8*T(5)*T(6)^5-T(2)^7*T(5)^2*T(6)^5+2*T(2)^8*T(6)^6-2*T(2)^7*T(5)*T(6)^6-T(2)^7*T(6)^7-T(2)^13+T(2)^12*T(5)*z+T(2)^11*T(5)^2-2*T(2)^8*T(5)^5+2*T(2)^7*T(5)^6*z+2*T(2)^6*T(5)^7+T(2)^12*T(6)*z+2*T(2)^11*T(5)*T(6)+2*T(2)^7*T(5)^5*T(6)*z-T(2)^6*T(5)^6*T(6)+T(2)^11*T(6)^2+2*T(2)^6*T(5)^5*T(6)^2-2*T(2)^8*T(6)^5+2*T(2)^7*T(5)*T(6)^5*z+2*T(2)^6*T(5)^2*T(6)^5+2*T(2)^7*T(6)^6*z-T(2)^6*T(5)*T(6)^6+2*T(2)^6*T(6)^7+T(2)^12*z^4+2*T(2)^12*z^2+2*T(2)^12*z-T(2)^12+2*T(2)^11*T(5)*z+2*T(2)^10*T(5)^2-T(2)^7*T(5)^5*z^2-T(2)^7*T(5)^5*z-2*T(2)^7*T(5)^5-T(2)^6*T(5)^6*z-T(2)^5*T(5)^7+2*T(2)^11*T(6)*z-T(2)^10*T(5)*T(6)-T(2)^6*T(5)^5*T(6)*z-2*T(2)^5*T(5)^6*T(6)+2*T(2)^10*T(6)^2-T(2)^5*T(5)^5*T(6)^2-T(2)^7*T(6)^5*z^2-T(2)^7*T(6)^5*z-2*T(2)^7*T(6)^5-T(2)^6*T(5)*T(6)^5*z-T(2)^5*T(5)^2*T(6)^5-T(2)^6*T(6)^6*z-2*T(2)^5*T(5)*T(6)^6-T(2)^5*T(6)^7+T(2)^11*z^5-2*T(2)^11*z^4+2*T(2)^11*z^2+2*T(2)^11*z-2*T(2)^11+T(2)^10*T(5)*z^4+2*T(2)^10*T(5)*z-2*T(2)^10*T(5)-T(2)^6*T(5)^5*z^2-T(2)^6*T(5)^5*z-2*T(2)^6*T(5)^5-T(2)^5*T(5)^6*z+T(2)^5*T(5)^6+T(2)^10*T(6)*z^4+2*T(2)^10*T(6)*z-2*T(2)^10*T(6)-T(2)^5*T(5)^5*T(6)*z+T(2)^5*T(5)^5*T(6)-T(2)^6*T(6)^5*z^2-T(2)^6*T(6)^5*z-2*T(2)^6*T(6)^5-T(2)^5*T(5)*T(6)^5*z+T(2)^5*T(5)*T(6)^5-T(2)^5*T(6)^6*z+T(2)^5*T(6)^6-2*T(2)^10*z^5+2*T(2)^10*z^4+T(2)^10*z^3-2*T(2)^10*z^2+T(2)^10*z-2*T(2)^10-2*T(2)^9*T(5)*z^5+T(2)^9*T(5)+T(2)^8*T(5)^2*z^4+2*T(2)^5*T(5)^5*z^3+T(2)^5*T(5)^5*z^2+2*T(2)^5*T(5)^5*z-T(2)^5*T(5)^5-2*T(2)^4*T(5)^6-2*T(2)^9*T(6)*z^5+T(2)^9*T(6)+2*T(2)^8*T(5)*T(6)*z^4-2*T(2)^4*T(5)^5*T(6)+T(2)^8*T(6)^2*z^4+2*T(2)^5*T(6)^5*z^3+T(2)^5*T(6)^5*z^2+2*T(2)^5*T(6)^5*z-T(2)^5*T(6)^5-2*T(2)^4*T(5)*T(6)^5-2*T(2)^4*T(6)^6+T(2)^9*z^6-T(2)^9*z^5-T(2)^9*z+T(2)^9+T(2)^8*T(5)*z^5+2*T(2)^8*T(5)*z^4+T(2)^8*T(5)+T(2)^7*T(5)^2*z^5+2*T(2)^7*T(5)^2*z^4+2*T(2)^7*T(5)^2+T(2)^6*T(5)^3*z^4+2*T(2)^4*T(5)^5*z-2*T(2)^4*T(5)^5-2*T(2)^3*T(5)^6+T(2)^2*T(5)^7+T(2)^8*T(6)*z^5+2*T(2)^8*T(6)*z^4+T(2)^8*T(6)+2*T(2)^7*T(5)*T(6)*z^5-T(2)^7*T(5)*T(6)*z^4-T(2)^7*T(5)*T(6)-2*T(2)^6*T(5)^2*T(6)*z^4-2*T(2)^3*T(5)^5*T(6)+2*T(2)^2*T(5)^6*T(6)+T(2)^7*T(6)^2*z^5+2*T(2)^7*T(6)^2*z^4+2*T(2)^7*T(6)^2-2*T(2)^6*T(5)*T(6)^2*z^4+T(2)^2*T(5)^5*T(6)^2+T(2)^6*T(6)^3*z^4+2*T(2)^4*T(6)^5*z-2*T(2)^4*T(6)^5-2*T(2)^3*T(5)*T(6)^5+T(2)^2*T(5)^2*T(6)^5-2*T(2)^3*T(6)^6+2*T(2)^2*T(5)*T(6)^6+T(2)^2*T(6)^7+T(2)^8*z^5+T(2)^8*z^4-T(2)^8+T(2)^7*T(5)*z^6-2*T(2)^7*T(5)*z^5-T(2)^7*T(5)*z^4+T(2)^7*T(5)*z+T(2)^6*T(5)^2*z^4+T(2)^6*T(5)^2-T(2)^5*T(5)^3*z^4+T(2)^4*T(5)^4*z^4+2*T(2)^3*T(5)^5-2*T(2)^2*T(5)^6*z-2*T(2)*T(5)^7+T(2)^7*T(6)*z^6-2*T(2)^7*T(6)*z^5-T(2)^7*T(6)*z^4+T(2)^7*T(6)*z+2*T(2)^6*T(5)*T(6)*z^4+2*T(2)^6*T(5)*T(6)+2*T(2)^5*T(5)^2*T(6)*z^4-T(2)^4*T(5)^3*T(6)*z^4-2*T(2)^2*T(5)^5*T(6)*z+T(2)*T(5)^6*T(6)+T(2)^6*T(6)^2*z^4+T(2)^6*T(6)^2+2*T(2)^5*T(5)*T(6)^2*z^4+T(2)^4*T(5)^2*T(6)^2*z^4-2*T(2)*T(5)^5*T(6)^2-T(2)^5*T(6)^3*z^4-T(2)^4*T(5)*T(6)^3*z^4+T(2)^4*T(6)^4*z^4+2*T(2)^3*T(6)^5-2*T(2)^2*T(5)*T(6)^5*z-2*T(2)*T(5)^2*T(6)^5-2*T(2)^2*T(6)^6*z+T(2)*T(5)*T(6)^6-2*T(2)*T(6)^7-2*T(2)^7*z^7-2*T(2)^7*z^6+T(2)^7*z^5+2*T(2)^7*z^2+2*T(2)^7*z-T(2)^7-2*T(2)^6*T(5)*z^6-T(2)^6*T(5)*z^5-T(2)^6*T(5)*z^4+2*T(2)^6*T(5)*z+2*T(2)^5*T(5)^2*z^6+2*T(2)^5*T(5)^2*z^5+T(2)^5*T(5)^2*z^4+2*T(2)^5*T(5)^2+T(2)^4*T(5)^3*z^5-T(2)^4*T(5)^3*z^4+T(2)^3*T(5)^4*z^4+T(2)^2*T(5)^5*z^2+T(2)^2*T(5)^5*z+2*T(2)^2*T(5)^5+T(2)*T(5)^6*z+T(5)^7-2*T(2)^6*T(6)*z^6-T(2)^6*T(6)*z^5-T(2)^6*T(6)*z^4+2*T(2)^6*T(6)*z-T(2)^5*T(5)*T(6)*z^6-T(2)^5*T(5)*T(6)*z^5+2*T(2)^5*T(5)*T(6)*z^4-T(2)^5*T(5)*T(6)-2*T(2)^4*T(5)^2*T(6)*z^5+2*T(2)^4*T(5)^2*T(6)*z^4-T(2)^3*T(5)^3*T(6)*z^4+T(2)*T(5)^5*T(6)*z+2*T(5)^6*T(6)+2*T(2)^5*T(6)^2*z^6+2*T(2)^5*T(6)^2*z^5+T(2)^5*T(6)^2*z^4+2*T(2)^5*T(6)^2-2*T(2)^4*T(5)*T(6)^2*z^5+2*T(2)^4*T(5)*T(6)^2*z^4+T(2)^3*T(5)^2*T(6)^2*z^4+T(5)^5*T(6)^2+T(2)^4*T(6)^3*z^5-T(2)^4*T(6)^3*z^4-T(2)^3*T(5)*T(6)^3*z^4+T(2)^3*T(6)^4*z^4+T(2)^2*T(6)^5*z^2+T(2)^2*T(6)^5*z+2*T(2)^2*T(6)^5+T(2)*T(5)*T(6)^5*z+T(5)^2*T(6)^5+T(2)*T(6)^6*z+2*T(5)*T(6)^6+T(6)^7-T(2)^6*z^6-2*T(2)^6*z^5-2*T(2)^6*z^4+2*T(2)^6*z^2+2*T(2)^6*z+T(2)^6+T(2)^5*T(5)*z^7-T(2)^5*T(5)*z^6-2*T(2)^5*T(5)*z^4+2*T(2)^5*T(5)*z-2*T(2)^5*T(5)+T(2)^4*T(5)^2*z^6-2*T(2)^4*T(5)^2*z^5+T(2)^4*T(5)^2*z^4+T(2)^3*T(5)^3*z^6-T(2)^3*T(5)^3*z^4+T(2)^2*T(5)^4*z^4+T(2)*T(5)^5*z^2+T(2)*T(5)^5*z-2*T(2)*T(5)^5+T(5)^6*z-T(5)^6+T(2)^5*T(6)*z^7-T(2)^5*T(6)*z^6-2*T(2)^5*T(6)*z^4+2*T(2)^5*T(6)*z-2*T(2)^5*T(6)+2*T(2)^4*T(5)*T(6)*z^6+T(2)^4*T(5)*T(6)*z^5+2*T(2)^4*T(5)*T(6)*z^4-2*T(2)^3*T(5)^2*T(6)*z^6+2*T(2)^3*T(5)^2*T(6)*z^4-T(2)^2*T(5)^3*T(6)*z^4+T(5)^5*T(6)*z-T(5)^5*T(6)+T(2)^4*T(6)^2*z^6-2*T(2)^4*T(6)^2*z^5+T(2)^4*T(6)^2*z^4-2*T(2)^3*T(5)*T(6)^2*z^6+2*T(2)^3*T(5)*T(6)^2*z^4+T(2)^2*T(5)^2*T(6)^2*z^4+T(2)^3*T(6)^3*z^6-T(2)^3*T(6)^3*z^4-T(2)^2*T(5)*T(6)^3*z^4+T(2)^2*T(6)^4*z^4+T(2)*T(6)^5*z^2+T(2)*T(6)^5*z-2*T(2)*T(6)^5+T(5)*T(6)^5*z-T(5)*T(6)^5+T(6)^6*z-T(6)^6+2*T(2)^5*z^7+T(2)^5*z^5-T(2)^5*z^4+T(2)^5*z^3-2*T(2)^5*z^2+T(2)^5*z-2*T(2)^4*T(5)*z^7+2*T(2)^4*T(5)*z^6-T(2)^4*T(5)*z^5-T(2)^4*T(5)*z^4+2*T(2)^3*T(5)^2*z^7+2*T(2)^3*T(5)^2*z^6+2*T(2)^2*T(5)^3*z^6-T(2)^2*T(5)^3*z^5-T(2)^2*T(5)^3*z^4+T(2)*T(5)^4*z^4-2*T(5)^5*z^3-T(5)^5*z^2-2*T(5)^5*z-2*T(2)^4*T(6)*z^7+2*T(2)^4*T(6)*z^6-T(2)^4*T(6)*z^5-T(2)^4*T(6)*z^4-T(2)^3*T(5)*T(6)*z^7-T(2)^3*T(5)*T(6)*z^6+T(2)^2*T(5)^2*T(6)*z^6+2*T(2)^2*T(5)^2*T(6)*z^5+2*T(2)^2*T(5)^2*T(6)*z^4-T(2)*T(5)^3*T(6)*z^4+2*T(2)^3*T(6)^2*z^7+2*T(2)^3*T(6)^2*z^6+T(2)^2*T(5)*T(6)^2*z^6+2*T(2)^2*T(5)*T(6)^2*z^5+2*T(2)^2*T(5)*T(6)^2*z^4+T(2)*T(5)^2*T(6)^2*z^4+2*T(2)^2*T(6)^3*z^6-T(2)^2*T(6)^3*z^5-T(2)^2*T(6)^3*z^4-T(2)*T(5)*T(6)^3*z^4+T(2)*T(6)^4*z^4-2*T(6)^5*z^3-T(6)^5*z^2-2*T(6)^5*z-T(2)^4*z^8-2*T(2)^4*z^7+T(2)^4*z^6+T(2)^4*z^5+T(2)^4*z^4-2*T(2)^3*T(5)*z^8-T(2)^3*T(5)*z^7+T(2)^3*T(5)*z^6-2*T(2)^3*T(5)*z^5+2*T(2)^3*T(5)*z^4-2*T(2)^2*T(5)^2*z^7-2*T(2)^2*T(5)^2*z^6-T(2)^2*T(5)^2*z^5-T(2)^2*T(5)^2*z^4-2*T(2)*T(5)^3*z^6-2*T(2)*T(5)^3*z^5-2*T(2)*T(5)^3*z^4+T(5)^4*z^4-2*T(2)^3*T(6)*z^8-T(2)^3*T(6)*z^7+T(2)^3*T(6)*z^6-2*T(2)^3*T(6)*z^5+2*T(2)^3*T(6)*z^4+T(2)^2*T(5)*T(6)*z^7+T(2)^2*T(5)*T(6)*z^6-2*T(2)^2*T(5)*T(6)*z^5-2*T(2)^2*T(5)*T(6)*z^4-T(2)*T(5)^2*T(6)*z^6-T(2)*T(5)^2*T(6)*z^5-T(2)*T(5)^2*T(6)*z^4-T(5)^3*T(6)*z^4-2*T(2)^2*T(6)^2*z^7-2*T(2)^2*T(6)^2*z^6-T(2)^2*T(6)^2*z^5-T(2)^2*T(6)^2*z^4-T(2)*T(5)*T(6)^2*z^6-T(2)*T(5)*T(6)^2*z^5-T(2)*T(5)*T(6)^2*z^4+T(5)^2*T(6)^2*z^4-2*T(2)*T(6)^3*z^6-2*T(2)*T(6)^3*z^5-2*T(2)*T(6)^3*z^4-T(5)*T(6)^3*z^4+T(6)^4*z^4+T(2)^3*z^9-T(2)^3*z^8-2*T(2)^3*z^7-2*T(2)^2*T(5)*z^8-2*T(2)^2*T(5)*z^7-2*T(2)^2*T(5)*z^6-T(2)^2*T(5)*z^5-2*T(2)*T(5)^2*z^7-T(2)*T(5)^2*z^6-2*T(2)*T(5)^2*z^5-T(5)^3*z^6+2*T(5)^3*z^5-2*T(2)^2*T(6)*z^8-2*T(2)^2*T(6)*z^7-2*T(2)^2*T(6)*z^6-T(2)^2*T(6)*z^5+T(2)*T(5)*T(6)*z^7-2*T(2)*T(5)*T(6)*z^6+T(2)*T(5)*T(6)*z^5+2*T(5)^2*T(6)*z^6+T(5)^2*T(6)*z^5-2*T(2)*T(6)^2*z^7-T(2)*T(6)^2*z^6-2*T(2)*T(6)^2*z^5+2*T(5)*T(6)^2*z^6+T(5)*T(6)^2*z^5-T(6)^3*z^6+2*T(6)^3*z^5+T(2)^2*z^9+T(2)^2*z^8+2*T(2)^2*z^7-T(2)^2*z^6-T(2)^2*z^5-2*T(2)*T(5)*z^9-T(2)*T(5)*z^8-T(2)*T(5)*z^7-T(2)*T(5)*z^6+2*T(5)^2*z^7-2*T(5)^2*z^6+2*T(5)^2*z^5-2*T(2)*T(6)*z^9-T(2)*T(6)*z^8-T(2)*T(6)*z^7-T(2)*T(6)*z^6-T(5)*T(6)*z^7+T(5)*T(6)*z^6-T(5)*T(6)*z^5+2*T(6)^2*z^7-2*T(6)^2*z^6+2*T(6)^2*z^5+2*T(2)*z^10-T(2)*z^9+2*T(2)*z^7+2*T(2)*z^6+T(2)*z^5+2*T(5)*z^9+2*T(5)*z^6-2*T(5)*z^5+2*T(6)*z^9+2*T(6)*z^6-2*T(6)*z^5+z^11-z^10-z^9-z^8-2*z^7+z^6 > option(redSB); > ideal j=std(norid);j; j[1]=T(5)^2*z^4+T(5)^2*z^3+T(5)^2*z^2-2*T(5)^2*z+2*T(2)*T(6)*z^4+2*T(2)*T(6)*z^3+T(2)*T(6)*z^2+2*T(2)*T(6)*z-T(2)*T(6)-2*T(5)*T(6)*z^4+T(5)*T(6)*z^3-2*T(5)*T(6)*z^2+2*T(5)*T(6)*z+2*T(6)^2*z^2-T(6)^2*z+2*T(6)^2-2*T(2)*z^5-T(2)*z^3+2*T(2)*z^2-T(2)*z+T(2)+2*T(5)*z^5+T(5)*z^4-T(5)*z^3-2*T(5)*z^2+T(5)*z-T(5)-T(6)*z^4-T(6)*z^3+2*T(6)*z-T(6)+2*z^6-2*z^5+z^4-2*z^3-2*z^2+z-2 j[2]=T(2)*T(5)-2*T(5)^2*z^3-2*T(5)^2*z^2-2*T(5)^2*z+T(2)*T(6)*z^3+T(2)*T(6)*z^2-2*T(2)*T(6)*z-2*T(2)*T(6)-T(5)*T(6)*z^3-2*T(5)*T(6)*z^2-T(5)*T(6)*z-T(5)*T(6)+T(6)^2*z+2*T(6)^2-T(2)*z^4+2*T(2)*z^2-T(2)*z+T(2)+T(5)*z^4-2*T(5)*z^3+2*T(5)*z^2+T(5)*z+2*T(6)*z^3+2*T(6)*z^2-2*T(6)+z^5-z^4-2*z^3+z^2+2*z j[3]=T(2)^2-T(5)^2*z^3+2*T(5)^2*z^2+2*T(5)^2*z-2*T(5)^2-2*T(2)*T(6)*z^3-T(2)*T(6)*z^2+2*T(2)*T(6)+2*T(5)*T(6)*z^3-2*T(5)*T(6)*z^2+T(5)*T(6)-2*T(6)^2*z+2*T(6)^2+2*T(2)*z^4-T(2)*z^3+T(2)*z^2-T(2)*z-2*T(2)-2*T(5)*z^4-T(5)*z^2-T(5)+T(6)*z^3-2*T(6)*z^2+2*T(6)*z-2*z^5-2*z^4-2*z^3+z^2-z+2 j[4]=T(6)^3+T(5)^2*z^2-2*T(5)^2*z+T(5)^2+2*T(2)*T(6)*z^2-2*T(2)*T(6)*z+T(2)*T(6)-2*T(5)*T(6)*z^2-T(5)*T(6)*z+T(6)^2*z+T(6)^2+2*T(2)*z^3+T(2)*z^2-T(2)*z+2*T(5)*z^3+2*T(5)*z^2+T(5)*z+2*T(5)+T(6)*z-T(6)+2*z^4-z^3+z+1 j[5]=T(5)*T(6)^2-T(5)^2*z^3+T(5)^2*z^2+T(5)^2-2*T(2)*T(6)*z^3+2*T(2)*T(6)*z^2-T(2)*T(6)*z+2*T(5)*T(6)*z^3+2*T(5)*T(6)*z+2*T(5)*T(6)-2*T(6)^2*z+T(6)^2+2*T(2)*z^4+T(2)*z^3-T(2)*z^2+2*T(2)*z+2*T(2)-2*T(5)*z^4-2*T(5)*z^3+T(5)*z^2-2*T(5)*z-T(5)+T(6)*z^3-2*T(6)*z^2-T(6)*z+T(6)-2*z^5+z^4+z^3-z^2+z+1 j[6]=T(2)*T(6)^2+T(5)^2*z^2+2*T(5)^2+2*T(2)*T(6)*z^2-2*T(5)*T(6)*z^2+T(5)*T(6)*z+T(5)*T(6)-2*T(2)*z^3-2*T(2)*z^2+2*T(2)*z+T(2)+2*T(5)*z^3+T(5)*z^2+2*T(5)*z-2*T(5)-T(6)*z^2-T(6)+2*z^4-2*z^3+z j[7]=T(5)^2*T(6)-2*T(5)^2*z^3-2*T(5)^2*z^2+2*T(5)^2*z+2*T(5)^2+T(2)*T(6)*z^3+T(2)*T(6)*z^2-2*T(2)*T(6)-T(5)*T(6)*z^3-2*T(5)*T(6)*z^2-T(5)*T(6)*z+2*T(5)*T(6)+T(6)^2*z+2*T(6)^2-T(2)*z^4-T(2)*z^2+2*T(2)*z-T(2)+T(5)*z^4-2*T(5)*z^3+T(5)*z^2+2*T(5)*z-2*T(5)+2*T(6)*z^3+2*T(6)*z^2+T(6)+z^5-z^4-2*z^3-2*z^2+2 j[8]=T(5)^3+T(5)^2*z^3-2*T(5)^2*z^2-2*T(5)^2*z-2*T(5)^2+2*T(2)*T(6)*z^3+T(2)*T(6)*z^2-T(2)*T(6)*z+2*T(2)*T(6)-2*T(5)*T(6)*z^3+2*T(5)*T(6)*z^2+2*T(6)^2*z-2*T(6)^2-2*T(2)*z^4+T(2)*z^3-T(2)*z-2*T(2)+2*T(5)*z^4+T(5)*z^2-2*T(5)-T(6)*z^3+2*T(6)*z^2-T(6)*z-2*T(6)+2*z^5+2*z^4+2*z^3+2*z^2-z-2 \end{verbatim} \end{document}