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Magaard, Shaska, Shpectorov, and Völklein give tables of smooth Riemann surfaces of genus \( 3 \leq g \leq 10\) with automorphism groups \(G\) satisfying \( \# G > 4(g-1)\). Their list is based on a computer search by Breuer.
They list a genus 7 Riemann surface with automorphism group (32,11) in the GAP library of small groups. The quotient of this surface by its automorphism group has genus zero, and the quotient morphism is branched over three points with ramification indices (4,4,8).
We use Macaulay2 and Magma to compute equations of this curve.
Magma V2.21-10 Sat Mar 12 2016 13:41:55 on ace-math01 [Seed = 79048832]
Type ? for help. Type -D to quit.
> load "autcv10c.txt";
Loading "autcv10c.txt"
> G:=SmallGroup(32,11);
> G;
GrpPC : G of order 32 = 2^5
PC-Relations:
G.1^2 = G.4,
G.3^2 = G.5,
G.4^2 = G.5,
G.2^G.1 = G.2 * G.3,
G.3^G.1 = G.3 * G.5,
G.3^G.2 = G.3 * G.5
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,7,[4,4,8]);
Set seed to 0.
Character Table of Group G
--------------------------
----------------------------------------------------------
Class | 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Size | 1 1 2 4 1 1 2 2 2 2 2 4 4 4
Order | 1 2 2 2 4 4 4 4 4 4 4 4 8 8
----------------------------------------------------------
p = 2 1 1 1 1 2 2 3 2 3 3 3 2 5 6
----------------------------------------------------------
X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 + 1 1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1
X.3 + 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1
X.4 + 1 1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 1
X.5 0 1 1 -1 -1 -1 -1 -I 1 I I -I 1 -I I
X.6 0 1 1 -1 -1 -1 -1 I 1 -I -I I 1 I -I
X.7 0 1 1 -1 1 -1 -1 -I 1 I I -I -1 I -I
X.8 0 1 1 -1 1 -1 -1 I 1 -I -I I -1 -I I
X.9 + 2 2 -2 0 2 2 0 -2 0 0 0 0 0 0
X.10 + 2 2 2 0 -2 -2 0 -2 0 0 0 0 0 0
X.11 0 2 -2 0 0 -2*I 2*I-1-I 0 1-I-1+I 1+I 0 0 0
X.12 0 2 -2 0 0 2*I -2*I-1+I 0 1+I-1-I 1-I 0 0 0
X.13 0 2 -2 0 0 -2*I 2*I 1+I 0-1+I 1-I-1-I 0 0 0
X.14 0 2 -2 0 0 2*I -2*I 1-I 0-1-I 1+I-1+I 0 0 0
Explanation of Character Value Symbols
--------------------------------------
I = RootOfUnity(4)
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 1
Rep G.5
[3] Order 2 Length 2
Rep G.3 * G.4
[4] Order 2 Length 4
Rep G.2
[5] Order 4 Length 1
Rep G.4
[6] Order 4 Length 1
Rep G.4 * G.5
[7] Order 4 Length 2
Rep G.1 * G.2 * G.4 * G.5
[8] Order 4 Length 2
Rep G.3
[9] Order 4 Length 2
Rep G.1 * G.2 * G.5
[10] Order 4 Length 2
Rep G.1 * G.2
[11] Order 4 Length 2
Rep G.1 * G.2 * G.4
[12] Order 4 Length 4
Rep G.2 * G.4
[13] Order 8 Length 4
Rep G.1
[14] Order 8 Length 4
Rep G.1 * G.4
Surface kernel generators: [ G.2 * G.4, G.1 * G.2 * G.4, G.1 * G.3 * G.4 * G.5
]
Is hyperelliptic? false
Is cyclic trigonal? false
Multiplicities of irreducibles in relevant G-modules:
I_1 =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1 =[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1 ]
I_2 =[ 1, 1, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0 ]
S_2 =[ 1, 2, 1, 1, 1, 0, 2, 2, 2, 4, 0, 1, 1, 1 ]
H^0(C,2K)=[ 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1 ]
I_3 =[ 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 5, 6, 5, 6 ]
S_3 =[ 2, 2, 1, 0, 1, 1, 2, 1, 4, 2, 7, 9, 7, 8 ]
H^0(C,3K)=[ 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 3, 2, 2 ]
I2timesS1=[ 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 7, 8, 7, 8 ]
Is clearly not generated by quadrics? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 32 and degree 16
[
[-z^8 0 0 0 0 0 0]
[ 0 0 1 0 0 0 0]
[ 0 -z^8 0 0 0 0 0]
[ 0 0 0 0 -z^8 0 0]
[ 0 0 0 1 0 0 0]
[ 0 0 0 0 0 0 z^8]
[ 0 0 0 0 0 1 0],
[ 1 0 0 0 0 0 0]
[ 0 0 -z^8 0 0 0 0]
[ 0 z^8 0 0 0 0 0]
[ 0 0 0 0 1 0 0]
[ 0 0 0 1 0 0 0]
[ 0 0 0 0 0 0 1]
[ 0 0 0 0 0 1 0]
]
Matrix Surface Kernel Generators:
[
[ -1 0 0 0 0 0 0]
[ 0 0 -1 0 0 0 0]
[ 0 1 0 0 0 0 0]
[ 0 0 0 0 -z^8 0 0]
[ 0 0 0 -z^8 0 0 0]
[ 0 0 0 0 0 0 z^8]
[ 0 0 0 0 0 z^8 0],
[ z^8 0 0 0 0 0 0]
[ 0 1 0 0 0 0 0]
[ 0 0 z^8 0 0 0 0]
[ 0 0 0 -1 0 0 0]
[ 0 0 0 0 -z^8 0 0]
[ 0 0 0 0 0 -1 0]
[ 0 0 0 0 0 0 z^8],
[ z^8 0 0 0 0 0 0]
[ 0 0 1 0 0 0 0]
[ 0 z^8 0 0 0 0 0]
[ 0 0 0 0 -z^8 0 0]
[ 0 0 0 -1 0 0 0]
[ 0 0 0 0 0 0 z^8]
[ 0 0 0 0 0 -1 0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 1
Multiplicity 1
[
x_3*x_5 + x_4*x_6
]
I2 contains a 1-dimensional subspace of CharacterRow 2
Dimension 2
Multiplicity 2
[
x_0^2,
x_1*x_5 + z^8*x_2*x_6
]
I2 contains a 1-dimensional subspace of CharacterRow 7
Dimension 2
Multiplicity 2
[
x_1*x_4 + z^8*x_2*x_3,
x_5*x_6
]
I2 contains a 1-dimensional subspace of CharacterRow 8
Dimension 2
Multiplicity 2
[
x_1*x_2,
x_3*x_4
]
I2 contains a 2-dimensional subspace of CharacterRow 9
Dimension 4
Multiplicity 2
[
x_1*x_6,
x_2*x_5,
x_3*x_6,
x_4*x_5
]
I2 contains a 4-dimensional subspace of CharacterRow 10
Dimension 8
Multiplicity 4
[
x_1^2,
x_1*x_3,
x_2^2,
x_2*x_4,
x_3^2,
x_4^2,
x_5^2,
x_6^2
]
> GL7K:=Parent(MatrixGens[1]);
> MatrixG:=sub;
> FindParallelBases(MatrixG,[Q[5][1],Q[5][2]],[Q[5][3],Q[5][4]]);
[z^8*x_4*x_5]
[ x_3*x_6]
> FindParallelBases(MatrixG,[Q[6][1],Q[6][3]],[Q[6][2],Q[6][4]]);
[-z^8*x_2*x_4]
[ x_1*x_3]
> FindParallelBases(MatrixG,[Q[6][1],Q[6][3]],[Q[6][5],Q[6][6]]);
[ x_3^2]
[-x_4^2]
> FindParallelBases(MatrixG,[Q[6][1],Q[6][3]],[Q[6][7],Q[6][8]]);
[ x_5^2]
[-x_6^2]
The output above shows that the ideal contains quadrics from three isotypical subspaces of \(S_2\). Note that the power of z\(=\zeta_{32}\) in our equations is always a multiple of 8. Therefore in the sequel we reduce these to i\(= \zeta_{4}\).
The first isotypical subspace, which corresponds to the character \( \chi_1\) in the character table shown above, yields a polynomial of the form \[ x_3 x_5 + x_4 x_6 \]
The second isotypical subspace, which corresponds to the character \( \chi_2\) in the character table shown above, yields a polynomial of the form \[ c_1(x_0^2) + c_2(x_1 x_5 + i x_2 x_6) \] Assume that \( c_1 \) and \(c_2\) are nonzero. Then by scaling \( x_0\) and dividing, we may assume that \( c_1 = c_2 =1\).
The third isotypical subspace, which corresponds to the character \( \chi_7\) in the character table shown above, yields a polynomial of the form \[ c_3(x_1 x_4 + i x_2 x_3) + c_4(x_5 x_6). \] Assume that \( c_3 \) and \(c_4\) are nonzero. Then by scaling \( x_5,x_6\) and dividing, we may assume that \( c_3 = c_4 =1\).
The fourth isotypical subspace, which corresponds to the character \( \chi_8\) in the character table shown above, yields a polynomial of the form \[ c_5(x_1 x_2 ) + c_6(x_3 x_4). \] Assume that \( c_5 \) and \(c_6\) are nonzero. Then by scaling \( x_3,x_4\) and dividing, we may assume that \( c_5 = c_6 =1\).
The fifth isotypical subspace corresponds to the character \( \chi_{9}\) in the character table shown above. Note that the matrix surface kernel generators have a block diagonal form with blocks of size \(1 \times 1\), \(2 \times 2\), \(2 \times 2\), and \(2 \times 2\). We therefore let \( \operatorname{Span}\langle x_1 x_6, x_2 x_5 \rangle \) generate one copy of \(V_{9}\) and use the FindParallelBases function to find an ordered bases of \( \operatorname{Span}\langle x_3 x_6, x_4 x_5\rangle \) such that the action of \(G\) is given by the same matrices relative to both ordered bases. This yields the basis
[z^8*x_4*x_5]
[ x_3*x_6]
The sixth isotypical subspace corresponds to the character \( \chi_{10}\) in the character table shown above. From the block decomposition of the matrix surface kernal generators, we let \( \operatorname{Span}\langle x_1^2, x_2^2 \rangle \) generate one copy of \(V_{10}\) and use the FindParallelBases function to find ordered bases of \( \operatorname{Span}\langle x_1 x_3, x_2 x_4\rangle \) , \( \operatorname{Span}\langle x_3^2, x_4^2\rangle \) , and \( \operatorname{Span}\langle x_5^2, x_6^2\rangle \) such that the action of \(G\) is given by the same matrices relative to all four ordered bases. This yields the basis
[-z^8*x_2*x_4]
[ x_1*x_3],
[ x_3^2]
[-x_4^2],
[ x_5^2]
[-x_6^2]
c9*(x_1^2)+c10*(-z^8*x_2*x_4)+c11*(x_3^2)+c12*(x_5^2),
c9*(x_2^2)+c10*(x_1*x_3)+c11*(-x_4^2)+c12*(-x_6^2),
c13*(x_1^2)+c14*(-z^8*x_2*x_4)+c15*(x_3^2)+c16*(x_5^2),
c13*(x_2^2)+c14*(x_1*x_3)+c15*(-x_4^2)+c16*(-x_6^2)
> K<z_32>:=CyclotomicField(32);
> i:=z_32^8;
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c_8:=1;
> c_11:=1;
> c_12:=1;
> c_15:=1;
> c_16:=1;
> X:=Scheme(P6,[
> x_3*x_5 + x_4*x_6,
> x_0^2+x_1*x_5 + i*x_2*x_6,
> x_1*x_4 + i*x_2*x_3+x_5*x_6,
> x_1*x_2+x_3*x_4,
> x_1*x_6+c_8*(i*x_4*x_5),
> x_2*x_5+c_8*(x_3*x_6),
> x_1^2 +c_11*(x_3^2)+c_12*(x_5^2),
> x_2^2+c_11*(-x_4^2)+c_12*(-x_6^2),
> -i*x_2*x_4+c_15*(x_3^2)+c_16*(x_5^2),
> x_1*x_3+c_15*(-x_4^2)+c_16*(-x_6^2)
> ]);
> Dimension(X);
-1
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition, ReesAlgebra, TangentCone
i1 : loadPackage("Cyclotomic");
i2 : K=cyclotomicField(4);
i3 : i=K_0;
i4 : S=K[c_8,c_11,c_12,c_15,c_16,Degrees=>{0,0,0,0,0}]
o4 = S
o4 : PolynomialRing
i5 : T=S[x_0..x_6];
i6 : I=ideal({
x_3*x_5 + x_4*x_6,
x_0^2+x_1*x_5 + i*x_2*x_6,
x_1*x_4 + i*x_2*x_3+x_5*x_6,
x_1*x_2+x_3*x_4,
x_1*x_6+c_8*(i*x_4*x_5),
x_2*x_5+c_8*(x_3*x_6),
x_1^2 +c_11*(x_3^2)+c_12*(x_5^2),
x_2^2+c_11*(-x_4^2)+c_12*(-x_6^2),
-i*x_2*x_4+c_15*(x_3^2)+c_16*(x_5^2),
x_1*x_3+c_15*(-x_4^2)+c_16*(-x_6^2)
});
o6 : Ideal of T
i7 : L=flatten entries gens gb(I,DegreeLimit=>4);
i8 : tally apply(L, i -> degree i)
o8 = Tally{{2, 0} => 10}
{3, 0} => 11
o8 : Tally
i9 : for i from 0 to #L-1 do (if degree(L_i) == {3,0} then print toString(L_i) << endl)
(c_8+ww_4*c_15)*x_4^2*x_6+ww_4*c_16*x_6^3
c_11*x_3^2*x_6+c_8*x_2*x_4*x_6+(ww_4*c_8+c_12)*x_5^2*x_6
c_8*x_3^2*x_6-x_2*x_4*x_6
c_16*x_5^3+(ww_4*c_8-c_15)*x_3*x_4*x_6
c_15*x_4^2*x_5+x_1*x_4*x_6+c_16*x_5*x_6^2
c_11*x_4^2*x_5+ww_4*c_8*x_1*x_4*x_6+(ww_4*c_8+c_12)*x_5*x_6^2
x_1*x_4*x_5-ww_4*x_2*x_4*x_6+x_5^2*x_6
(c_11+ww_4)*x_3*x_4^2+(ww_4*c_8+c_12)*x_3*x_6^2
(c_11+ww_4)*x_1*x_4^2+ww_4*x_4*x_5*x_6+c_12*x_1*x_6^2
x_3^2*x_4+c_15*x_2*x_4^2+c_16*x_2*x_6^2
c_11*x_3^3+c_15*x_1*x_4^2-c_12*x_4*x_5*x_6+c_16*x_1*x_6^2
The leading coefficients of the generators above suggest the choices \( c_8 = -i c_{15}\) and \( c_{11} = -i\). We rerun the calculation with these choices.
i18 : S=K[c_12,c_15,c_16,Degrees=>{0,0,0}]
o18 = S
o18 : PolynomialRing
i19 : c_8=-i*c_15;
i20 : c_11=-i;
i21 : T=S[x_0..x_6];
i22 : I=ideal({
x_3*x_5 + x_4*x_6,
x_0^2+x_1*x_5 + i*x_2*x_6,
x_1*x_4 + i*x_2*x_3+x_5*x_6,
x_1*x_2+x_3*x_4,
x_1*x_6+c_8*(i*x_4*x_5),
x_2*x_5+c_8*(x_3*x_6),
x_1^2 +c_11*(x_3^2)+c_12*(x_5^2),
x_2^2+c_11*(-x_4^2)+c_12*(-x_6^2),
-i*x_2*x_4+c_15*(x_3^2)+c_16*(x_5^2),
x_1*x_3+c_15*(-x_4^2)+c_16*(-x_6^2)
});
o22 : Ideal of T
i23 : L=flatten entries gens gb(I);
i24 : tally apply(L, i -> degree i)
o24 = Tally{{2, 0} => 10}
{3, 0} => 23
{4, 0} => 12
o24 : Tally
i25 : for i from 0 to #L-1 do (if degree(L_i) == {3,0} then print toString(L_i) << endl)
c_16*x_6^3
c_16*x_5*x_6^2
(c_12*c_15-ww_4)*x_4*x_6^2
(c_12+c_15)*x_3*x_6^2
(c_12*c_15-ww_4)*x_1*x_6^2
c_16*x_5^2*x_6
x_4*x_5*x_6-ww_4*c_12*x_1*x_6^2
(c_12+c_15)*x_1*x_5*x_6
(c_15^2+ww_4)*x_4^2*x_6
(c_15^2+ww_4)*x_3*x_4*x_6
(c_15^2+ww_4)*x_2*x_4*x_6+(ww_4*c_12*c_15+ww_4*c_15^2)*x_5^2*x_6
(c_15^2+ww_4)*x_1*x_4*x_6+(c_12*c_15+c_15^2)*x_5*x_6^2
x_3^2*x_6+c_15*x_2*x_4*x_6+(ww_4*c_12+ww_4*c_15)*x_5^2*x_6
c_16*x_5^3
c_12*x_4*x_5^2-x_1*x_5*x_6
x_4^2*x_5+ww_4*c_15*x_1*x_4*x_6+(ww_4*c_12+ww_4*c_15)*x_5*x_6^2
x_1*x_4*x_5-ww_4*x_2*x_4*x_6+x_5^2*x_6
(c_15^2+ww_4)*x_4^3+(-c_12-c_15)*x_4*x_6^2
(c_15^2+ww_4)*x_3*x_4^2+c_16*x_1*x_5^2+c_15*c_16*x_3*x_6^2
(c_15^2+ww_4)*x_2*x_4^2-c_16*x_4*x_5^2+c_15*c_16*x_2*x_6^2
(c_15^2+ww_4)*x_1*x_4^2+(c_12*c_16+c_15*c_16)*x_1*x_6^2
x_3^2*x_4+c_15*x_2*x_4^2+c_16*x_2*x_6^2
x_3^3+ww_4*c_15*x_1*x_4^2+(c_12^2+ww_4*c_16)*x_1*x_6^2
The leading coefficients of the generators above suggest the choices \( c_{15}= -\zeta_{8}^3\) and \(c_{12} = -c_{15}\). We rerun the calculation with these choices.
i52 :
K=cyclotomicField(8);
i53 : z_8=K_0;
i54 : i=z_8^2;
i55 : S=K[c_16,Degrees=>{0}]
o55 = S
o55 : PolynomialRing
i56 : c_15=z_8^3;
i57 : c_8=-i*c_15;
i58 : c_11=-i;
i59 : c_12=-z_8^3;
i60 : T=S[x_0..x_6];
i61 : I=ideal({
x_3*x_5 + x_4*x_6,
x_0^2+x_1*x_5 + i*x_2*x_6,
x_1*x_4 + i*x_2*x_3+x_5*x_6,
x_1*x_2+x_3*x_4,
x_1*x_6+c_8*(i*x_4*x_5),
x_2*x_5+c_8*(x_3*x_6),
x_1^2 +c_11*(x_3^2)+c_12*(x_5^2),
x_2^2+c_11*(-x_4^2)+c_12*(-x_6^2),
-i*x_2*x_4+c_15*(x_3^2)+c_16*(x_5^2),
x_1*x_3+c_15*(-x_4^2)+c_16*(-x_6^2)
});
o61 : Ideal of T
i62 : L=flatten entries gens gb(I);
i63 : tally apply(L, i -> degree i)
o63 = Tally{{2, 0} => 10}
{3, 0} => 6
o63 : Tally
i64 : for i from 0 to #L-1 do (if degree(L_i) == {3,0} then print toString(L_i) << endl)
c_16*x_6^3
c_16*x_5*x_6^2
c_16*x_5^2*x_6
c_16*x_1*x_5*x_6-ww_8^2*c_16*x_2*x_6^2
c_16*x_5^3
c_16*x_1*x_5^2+ww_8^3*c_16*x_3*x_6^2
> K<z_8>:=CyclotomicField(8);
> i:=z_8^2;
> GL7K:=GeneralLinearGroup(7,K);
> A:=GL7K!Matrix([
> [-1,0,0,0,0,0,0],
> [0,0,-1,0,0,0,0],
> [0,1,0,0,0,0,0],
> [0,0,0,0,-i,0,0],
> [0,0,0,-i,0,0,0],
> [0,0,0,0,0,0,i],
> [0,0,0,0,0,i,0]
> ]);
> B:=GL7K!Matrix([
> [i,0,0,0,0,0,0],
> [0,1,0,0,0,0,0],
> [0,0,i,0,0,0,0],
> [0,0,0,-1,0,0,0],
> [0,0,0,0,-i,0,0],
> [0,0,0,0,0,-1,0],
> [0,0,0,0,0,0,i]
> ]);
> Order(A);
4
> Order(B);
4
> IdentifyGroup(sub<GL7K | A,B>);
<32, 11>
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[x_3*x_5+x_4*x_6,
> x_0^2+x_1*x_5+z_8^2*x_2*x_6,
> x_1*x_4+z_8^2*x_2*x_3+x_5*x_6,
> x_1*x_2+x_3*x_4,
> x_1*x_6+z_8^3*x_4*x_5,
> x_2*x_5+z_8*x_3*x_6,
> x_1^2-z_8^2*x_3^2-z_8^3*x_5^2,
> x_2^2+z_8^2*x_4^2+z_8^3*x_6^2,
> -z_8^2*x_2*x_4+z_8^3*x_3^2,
> x_1*x_3-z_8^3*x_4^2]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
-x_0
x_2
-x_1
-z_8^2*x_4
-z_8^2*x_3
z_8^2*x_6
z_8^2*x_5
and inverse
-x_0
-x_2
x_1
z_8^2*x_4
z_8^2*x_3
-z_8^2*x_6
-z_8^2*x_5
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
z_8^2*x_0
x_1
z_8^2*x_2
-x_3
-z_8^2*x_4
-x_5
z_8^2*x_6
and inverse
-z_8^2*x_0
x_1
-z_8^2*x_2
-x_3
z_8^2*x_4
-x_5
-z_8^2*x_6
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition, ReesAlgebra, TangentCone
i1 : loadPackage("Cyclotomic");
i2 : K=cyclotomicField(8);
i3 : z_8=K_0;
i4 : i=z_8^2;
i5 : R=K[x_0..x_6,MonomialOrder=>Lex];
i6 : I=ideal {
x_3*x_5+x_4*x_6,
x_0^2+x_1*x_5+z_8^2*x_2*x_6,
x_1*x_4+z_8^2*x_2*x_3+x_5*x_6,
x_1*x_2+x_3*x_4,
x_1*x_6+z_8^3*x_4*x_5,
x_2*x_5+z_8*x_3*x_6,
x_1^2-z_8^2*x_3^2-z_8^3*x_5^2,
x_2^2+z_8^2*x_4^2+z_8^3*x_6^2,
-z_8^2*x_2*x_4+z_8^3*x_3^2,
x_1*x_3-z_8^3*x_4^2
};
o6 : Ideal of R
i7 : betti res I
0 1 2 3 4 5
o7 = total: 1 10 25 25 10 1
0: 1 . . . . .
1: . 10 16 9 . .
2: . . 9 16 10 .
3: . . . . . 1
o7 : BettiTally