|
Magaard, Shaska, Shpectorov, and Völklein give tables of smooth Riemann surfaces of genus \( g \leq 10\) with large automorphism groups (that is, \( \# G > 4(g-1)\)). Their list is based on a computer search by Breuer.
Their tables list a genus 7 Riemann surface with automorphism group (144,127) 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 (2,3,12).
We use Macaulay2 and Magma to compute equations of this curve. The main tools are the Eichler trace formula; black-box commands in Magma for obtaining matrix generators of a representation of a finite group having a specified character; and a partial computation of a flattening stratification using Gröbner bases in Macaulay2
Magma V2.21-7 Wed May 25 2016 16:47:14 on Davids-MacBook-Pro-2 [Seed =
908553734]
+-------------------------------------------------------------------+
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| Simons Foundation |
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Type ? for help. Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> G:=SmallGroup(144,127);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,7,[2,3,12]);
Set seed to 0.
Character Table of Group G
--------------------------
-----------------------------------------------------------------------------
Class | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Size | 1 1 18 2 4 4 8 8 3 3 6 2 4 4 8 8 12
Order | 1 2 2 3 3 3 3 3 4 4 4 6 6 6 6 6 12
-----------------------------------------------------------------------------
p = 2 1 1 1 4 6 5 8 7 2 2 2 4 6 5 8 7 12
p = 3 1 2 3 1 1 1 1 1 10 9 11 2 2 2 2 2 11
-----------------------------------------------------------------------------
X.1 + 1 1 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 1 1 1
X.3 0 1 1 1 1 J -1-J J-1-J 1 1 1 1 J -1-J J-1-J 1
X.4 0 1 1 1 1 -1-J J-1-J J 1 1 1 1 -1-J J-1-J J 1
X.5 0 1 1 -1 1 -1-J J-1-J J -1 -1 1 1 -1-J J-1-J J 1
X.6 0 1 1 -1 1 J -1-J J-1-J -1 -1 1 1 J -1-J J-1-J 1
X.7 + 2 2 0 -1 2 2 -1 -1 0 0 2 -1 2 2 -1 -1 -1
X.8 0 2 2 0 -1 2*J-2-2*J -J 1+J 0 0 2 -1 2*J-2-2*J -J 1+J -1
X.9 0 2 2 0 -1-2-2*J 2*J 1+J -J 0 0 2 -1-2-2*J 2*J 1+J -J -1
X.10 0 2 -2 0 2 -1 -1 -1 -1 -2*I 2*I 0 -2 1 1 1 1 0
X.11 0 2 -2 0 2 -1 -1 -1 -1 2*I -2*I 0 -2 1 1 1 1 0
X.12 0 2 -2 0 2 -J 1+J -J 1+J 2*I -2*I 0 -2 J -1-J J-1-J 0
X.13 0 2 -2 0 2 1+J -J 1+J -J -2*I 2*I 0 -2 -1-J J-1-J J 0
X.14 0 2 -2 0 2 1+J -J 1+J -J 2*I -2*I 0 -2 -1-J J-1-J J 0
X.15 0 2 -2 0 2 -J 1+J -J 1+J -2*I 2*I 0 -2 J -1-J J-1-J 0
X.16 + 3 3 -1 3 0 0 0 0 3 3 -1 3 0 0 0 0 -1
X.17 + 3 3 1 3 0 0 0 0 -3 -3 -1 3 0 0 0 0 -1
X.18 + 4 -4 0 -2 -2 -2 1 1 0 0 0 2 2 2 -1 -1 0
X.19 0 4 -4 0 -2 -2*J 2+2*J J-1-J 0 0 0 2 2*J-2-2*J -J 1+J 0
X.20 0 4 -4 0 -2 2+2*J -2*J-1-J J 0 0 0 2-2-2*J 2*J 1+J -J 0
X.21 + 6 6 0 -3 0 0 0 0 0 0 -2 -3 0 0 0 0 1
-----------------------------
Class | 18 19 20 21
Size | 12 12 12 12
Order | 12 12 12 12
-----------------------------
p = 2 13 13 14 14
p = 3 9 10 9 10
-----------------------------
X.1 + 1 1 1 1
X.2 + -1 -1 -1 -1
X.3 0 -1-J -1-J J J
X.4 0 J J -1-J -1-J
X.5 0 -J -J 1+J 1+J
X.6 0 1+J 1+J -J -J
X.7 + 0 0 0 0
X.8 0 0 0 0 0
X.9 0 0 0 0 0
X.10 0 -I I -I I
X.11 0 I -I I -I
X.12 0 Z1 -Z1 Z1#5-Z1#5
X.13 0 -Z1#5 Z1#5 -Z1 Z1
X.14 0 Z1#5-Z1#5 Z1 -Z1
X.15 0 -Z1 Z1-Z1#5 Z1#5
X.16 + 0 0 0 0
X.17 + 0 0 0 0
X.18 + 0 0 0 0
X.19 0 0 0 0 0
X.20 0 0 0 0 0
X.21 + 0 0 0 0
Explanation of Character Value Symbols
--------------------------------------
# denotes algebraic conjugation, that is,
#k indicates replacing the root of unity w by w^k
J = RootOfUnity(3)
I = RootOfUnity(4)
Z1 = (CyclotomicField(12: Sparse := true)) ! [ RationalField() | 0, -1, 0,
-1 ]
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 1
Rep G.6
[3] Order 2 Length 18
Rep G.1 * G.3
[4] Order 3 Length 2
Rep G.5
[5] Order 3 Length 4
Rep G.2^2
[6] Order 3 Length 4
Rep G.2
[7] Order 3 Length 8
Rep G.2^2 * G.5
[8] Order 3 Length 8
Rep G.2 * G.5
[9] Order 4 Length 3
Rep G.1 * G.6
[10] Order 4 Length 3
Rep G.1
[11] Order 4 Length 6
Rep G.3
[12] Order 6 Length 2
Rep G.5 * G.6
[13] Order 6 Length 4
Rep G.2^2 * G.6
[14] Order 6 Length 4
Rep G.2 * G.6
[15] Order 6 Length 8
Rep G.2^2 * G.5 * G.6
[16] Order 6 Length 8
Rep G.2 * G.5 * G.6
[17] Order 12 Length 12
Rep G.3 * G.5
[18] Order 12 Length 12
Rep G.1 * G.2
[19] Order 12 Length 12
Rep G.1 * G.2 * G.6
[20] Order 12 Length 12
Rep G.1 * G.2^2
[21] Order 12 Length 12
Rep G.1 * G.2^2 * G.6
Surface kernel generators: [ G.1 * G.4 * G.6, G.2^2 * G.4 * G.5, G.1 * G.2 *
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, 0, 0, 0, 0, 0, 0, 0 ]
S_1 =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ]
I_2 =[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ]
S_2 =[ 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 1, 2 ]
H^0(C,2K)=[ 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1 ]
I_3 =[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 1, 1, 0, 3, 2, 2, 1 ]
S_3 =[ 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 3, 3, 3, 1, 2, 0, 4, 3, 3, 2 ]
H^0(C,3K)=[ 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1 ]
I2timesS1=[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 1, 1, 0, 4, 3, 3, 1 ]
Is clearly not generated by quadrics? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 144 and degree 48
[
[ -1 0 0 0 0 0 0]
[ 0 z^36 0 0 0 0 0]
[ 0 0 z^36 0 0 0 0]
[ 0 0 0 0 -1 z^24 - 1 0]
[ 0 0 0 0 0 0 1]
[ 0 0 0 z^24 0 0 -z^24]
[ 0 0 0 0 -1 0 0],
[ -z^24 0 0 0 0 0 0]
[ 0 -1 z^24 0 0 0 0]
[ 0 z^24 - 1 0 0 0 0 0]
[ 0 0 0 z^24 - 1 0 0 0]
[ 0 0 0 0 z^24 z^24 0]
[ 0 0 0 0 -z^24 0 0]
[ 0 0 0 z^24 - 1 0 0 1],
[ 1 0 0 0 0 0 0]
[ 0 -z^24 + 1 -z^24 0 0 0 0]
[ 0 -z^24 z^24 - 1 0 0 0 0]
[ 0 0 0 z^24 0 0 -z^24]
[ 0 0 0 0 -z^24 + 1 -z^24 + 1 0]
[ 0 0 0 0 -1 z^24 - 1 0]
[ 0 0 0 1 0 0 -z^24],
[ 1 0 0 0 0 0 0]
[ 0 z^24 -z^24 + 1 0 0 0 0]
[ 0 -z^24 + 1 -z^24 0 0 0 0]
[ 0 0 0 z^24 - 1 0 0 1]
[ 0 0 0 0 0 z^24 0]
[ 0 0 0 0 z^24 - 1 0 0]
[ 0 0 0 z^24 - 1 0 0 -z^24 + 1],
[ 1 0 0 0 0 0 0]
[ 0 1 0 0 0 0 0]
[ 0 0 1 0 0 0 0]
[ 0 0 0 z^24 - 1 0 0 0]
[ 0 0 0 0 -z^24 0 0]
[ 0 0 0 0 0 -z^24 0]
[ 0 0 0 0 0 0 z^24 - 1],
[ 1 0 0 0 0 0 0]
[ 0 -1 0 0 0 0 0]
[ 0 0 -1 0 0 0 0]
[ 0 0 0 -1 0 0 0]
[ 0 0 0 0 -1 0 0]
[ 0 0 0 0 0 -1 0]
[ 0 0 0 0 0 0 -1]
]
Matrix Surface Kernel Generators:
[
[-1 0 0 0 0 0 0]
[0 -z^36 + z^12 -z^12 0 0 0 0]
[0 -z^12 z^36 - z^12 0 0 0 0]
[0 0 0 0 z^24 z^24 0]
[0 0 0 -z^24 + 1 0 0 z^24 - 1]
[0 0 0 0 0 0 -z^24 + 1]
[0 0 0 0 0 z^24 0],
[ z^24 - 1 0 0 0 0 0 0]
[ 0 -1 z^24 - 1 0 0 0 0]
[ 0 z^24 0 0 0 0 0]
[ 0 0 0 z^24 - 1 0 0 1]
[ 0 0 0 0 z^24 - 1 0 0]
[ 0 0 0 0 -z^24 + 1 -z^24 0]
[ 0 0 0 0 0 0 1],
[z^24 0 0 0 0 0 0]
[0 z^36 - z^12 z^36 0 0 0 0]
[0 0 z^12 0 0 0 0]
[0 0 0 0 -z^24 + 1 0 0]
[0 0 0 -1 0 0 1]
[0 0 0 z^24 0 0 0]
[0 0 0 0 0 z^24 0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 4
Dimension 2
Multiplicity 2
[
x_0^2,
x_3*x_4 - z^24*x_3*x_5 - z^24*x_5*x_6
]
I2 contains a 3-dimensional subspace of CharacterRow 17
Dimension 6
Multiplicity 2
[
x_1^2,
x_1*x_2,
x_2^2,
x_3*x_4 + z^24*x_5*x_6,
x_3*x_5,
x_4*x_6 - z^24*x_5*x_6
]
I2 contains a 6-dimensional subspace of CharacterRow 21
Dimension 12
Multiplicity 2
[
x_1*x_3 - z^24*x_2*x_6,
x_1*x_4 + (-z^24 + 1)*x_2*x_5,
x_1*x_5 - x_2*x_5,
x_1*x_6 + x_2*x_6,
x_2*x_3 - z^24*x_2*x_6,
x_2*x_4 + (-z^24 - 1)*x_2*x_5,
x_3^2,
x_3*x_6,
x_4^2,
x_4*x_5,
x_5^2,
x_6^2
]
This shows that the ideal contains quadrics from three isotypical subspaces of \(S_2\). Note that the power of z\(=\zeta_{144}\) in our equations is always a multiple of 12. Therefore in the sequel we reduce these to z_12\(= \zeta_{12}\).
The first isotypical subspace, which corresponds to the character \( \chi_4\) in the character table shown above, yields a polynomial of the form \[ c_1(x_0^2) + c_2(x_3 x_6 - \zeta_{12}^{2} x_4 x_5 - \zeta_{12}^{2}x_4 x_6) \] After scaling \(x_0\) we may assume that \(c_1= c_2 = 1\).
The second isotypical subspace corresponds to the character \( \chi_{17}\) in the character table shown above. Note that the matrices generating the action have a block form with blocks of sizes \(1 \times 1\), \(2 \times 2\), and \( 4 \times 4\). We therefore let \( \operatorname{Span}\langle x_1^2,x_1 x_2,x_2^2\rangle \) generate one copy of \(V_{17}\) and use the FindParallelBases function to find an ordered basis of the span of the remaining three polynomials such that the action of \(G\) is given by the same matrices relative to both ordered bases.
> GL7K:=Parent(MatrixGens[1]);
> MatrixG:=sub<GL7K | MatrixGens>;
> FindParallelBases(MatrixG,[Q[2][1],Q[2][2],Q[2][3]],[Q[2][4],Q[2][5],Q[2][6]\
]);
[ x_3*x_4 + z^24*x_3*x_5 + 2*x_4*x_6 - z^24*x_5*x_6]
[ (-2*z^24 + 1)*x_3*x_4 + z^24*x_3*x_5 - 2*z^24*x_4*x_6 + z^24*x_5*x_6]
[-x_3*x_4 + (-z^24 + 2)*x_3*x_5 + (2*z^24 - 2)*x_4*x_6 + (-z^24 + 2)*x_5*x_6]c_3*(x_1^2) +c_4*(x_3*x_4 + z_12^2*x_3*x_5 + 2*x_4*x_6 - z_12^2*x_5*x_6),
c_3*(x_1*x_2) + c_4*((-2*z_12^2 + 1)*x_3*x_4 + z_12^2*x_3*x_5 - 2*z_12^2*x_4*x_6 + z_12^2*x_5*x_6),
c_3*(x_2^2) + c_4*(-x_3*x_4 + (-z_12^2 + 2)*x_3*x_5 + (2*z_12^2 - 2)*x_4*x_6 + (-z_12^2 + 2)*x_5*x_6)Similarly, we compute parallel bases for the third isotypical subspace, which corresponds to the character \( \chi_{24}\), and get candidate polynomials for this isotypical subspace.
For generic values of \(c_6\), the intersection of these 10 quadrics in \(\mathbb{P}^6\) is empty. Here is an example showing this:> K<z_12>:=CyclotomicField(12);
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c_6:=1;
> X:=Scheme(P6,[
> x_0^2 + x_3*x_4 - z_12^2*x_3*x_5 - z_12^2*x_5*x_6,
> x_1^2 +x_3*x_4 + z_12^2*x_3*x_5 + 2*x_4*x_6 - z_12^2*x_5*x_6,
> x_1*x_2 + (-2*z_12^2 + 1)*x_3*x_4 + z_12^2*x_3*x_5 - 2*z_12^2*x_4*x_6 + z_12\
^2*x_5*x_6,
> x_2^2 + -x_3*x_4 + (-z_12^2 + 2)*x_3*x_5 + (2*z_12^2 - 2)*x_4*x_6 + (-z_12^2\
+ 2)*x_5*x_6,
> x_1*x_3 - z_12^2*x_2*x_6 + c_6*(z_12*x_4^2 + (z_12^3 - 2*z_12)*x_4*x_5 + z_1\
2*x_5^2),
> x_1*x_4 + (-z_12^2 + 1)*x_2*x_5 + c_6*(-x_3*x_6 - x_6^2),
> x_1*x_5 - x_2*x_5 + c_6*(x_3^2 + (-z_12^2 + 2)*x_3*x_6),
> x_1*x_6 + x_2*x_6 + c_6*(-z_12*x_4^2 + z_12*x_4*x_5 - z_12*x_5^2),
> x_2*x_3 - z_12^2*x_2*x_6 + c_6*(z_12*x_4^2),
> x_2*x_4 + (-z_12^2 - 1)*x_2*x_5 + c_6*(z_12^2*x_3^2 + 2*z_12^2*x_3*x_6 + z_1\
2^2*x_6^2)
> ]);
> Dimension(X);
-1Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition, ReesAlgebra, TangentCone
i1 : loadPackage("Cyclotomic");
i2 : K=cyclotomicField(12);
i3 : z_12=K_0;
i4 : S=K[c_6,Degrees=>{0}];
i5 : T=S[x_0..x_6];
i6 : I=ideal({
x_0^2 + x_3*x_4 - z_12^2*x_3*x_5 - z_12^2*x_5*x_6,
x_1^2 +x_3*x_4 + z_12^2*x_3*x_5 + 2*x_4*x_6 - z_12^2*x_5*x_6,
x_1*x_2 + (-2*z_12^2 + 1)*x_3*x_4 + z_12^2*x_3*x_5 - 2*z_12^2*x_4*x_6 + z_12^2*x_5*x_6,
x_2^2 + -x_3*x_4 + (-z_12^2 + 2)*x_3*x_5 + (2*z_12^2 - 2)*x_4*x_6 + (-z_12^2 + 2)*x_5*x_6,
x_1*x_3 - z_12^2*x_2*x_6 + c_6*(z_12*x_4^2 + (z_12^3 - 2*z_12)*x_4*x_5 + z_12*x_5^2),
x_1*x_4 + (-z_12^2 + 1)*x_2*x_5 + c_6*(-x_3*x_6 - x_6^2),
x_1*x_5 - x_2*x_5 + c_6*(x_3^2 + (-z_12^2 + 2)*x_3*x_6),
x_1*x_6 + x_2*x_6 + c_6*(-z_12*x_4^2 + z_12*x_4*x_5 - z_12*x_5^2),
x_2*x_3 - z_12^2*x_2*x_6 + c_6*(z_12*x_4^2),
x_2*x_4 + (-z_12^2 - 1)*x_2*x_5 + c_6*(z_12^2*x_3^2 + 2*z_12^2*x_3*x_6 + z_12^2*x_6^2)
});
o6 : Ideal of T
i7 : L=flatten entries gens gb(I);
i8 : L3=select(L, i -> degree i == {3,0});
i9 : L3c=unique apply(L3, i -> leadCoefficient i);
i10 : for i from 0 to #L3c-1 do (print toString(L3c_i) << endl)
c_6^2-2*ww_12^3
1
> K<z_12>:=CyclotomicField(12);
> z_6:=z_12^2;
> z_3:=z_12^4;
> c_6:=z_12^3+1;
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[
> x_0^2 + x_3*x_4 - z_12^2*x_3*x_5 - z_12^2*x_5*x_6,
> x_1^2 +x_3*x_4 + z_12^2*x_3*x_5 + 2*x_4*x_6 - z_12^2*x_5*x_6,
> x_1*x_2 + (-2*z_12^2 + 1)*x_3*x_4 + z_12^2*x_3*x_5 - 2*z_12^2*x_4*x_6 + z_12\
^2*x_5*x_6,
> x_2^2 + -x_3*x_4 + (-z_12^2 + 2)*x_3*x_5 + (2*z_12^2 - 2)*x_4*x_6 + (-z_12^2\
+ 2)*x_5*x_6,
> x_1*x_3 - z_12^2*x_2*x_6 + c_6*(z_12*x_4^2 + (z_12^3 - 2*z_12)*x_4*x_5 + z_1\
2*x_5^2),
> x_1*x_4 + (-z_12^2 + 1)*x_2*x_5 + c_6*(-x_3*x_6 - x_6^2),
> x_1*x_5 - x_2*x_5 + c_6*(x_3^2 + (-z_12^2 + 2)*x_3*x_6),
> x_1*x_6 + x_2*x_6 + c_6*(-z_12*x_4^2 + z_12*x_4*x_5 - z_12*x_5^2),
> x_2*x_3 - z_12^2*x_2*x_6 + c_6*(z_12*x_4^2),
> x_2*x_4 + (-z_12^2 - 1)*x_2*x_5 + c_6*(z_12^2*x_3^2 + 2*z_12^2*x_3*x_6 + z_1\
2^2*x_6^2)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> A:=Matrix([
> [-1,0,0,0,0,0,0],
> [0,z_12^-1,-z_12, 0,0,0,0],
> [0,-z_12,-z_12^-1,0,0,0,0],
> [0,0,0,0,z_12^2,z_12^2,0],
> [0,0,0,-z_3,0,0,z_3],
> [0,0,0,0,0,0,-z_3],
> [0,0,0,0,0,z_6,0]
> ]);
> B:=Matrix([
> [z_3,0,0,0,0,0,0],
> [0,-1,z_3,0,0,0,0],
> [0,z_6,0,0,0,0,0],
> [0,0,0,z_3,0,0,1],
> [0,0,0,0,z_3,0,0],
> [0,0,0,0,-z_3,-z_6,0],
> [0,0,0,0,0,0,1]
> ]);
> Order(A);
2
> Order(B);
3
> Order( (A*B)^-1);
12
> GL7K:=GeneralLinearGroup(7,K);
> IdentifyGroup(sub<GL7K | A,B>);
<144, 127>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
-x_0
(-z_12^3 + z_12)*x_1 - z_12*x_2
-z_12*x_1 + (z_12^3 - z_12)*x_2
(-z_12^2 + 1)*x_4
z_12^2*x_3
z_12^2*x_3 + z_12^2*x_6
(z_12^2 - 1)*x_4 + (-z_12^2 + 1)*x_5
and inverse
-x_0
(-z_12^3 + z_12)*x_1 - z_12*x_2
-z_12*x_1 + (z_12^3 - z_12)*x_2
(-z_12^2 + 1)*x_4
z_12^2*x_3
z_12^2*x_3 + z_12^2*x_6
(z_12^2 - 1)*x_4 + (-z_12^2 + 1)*x_5
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
(z_12^2 - 1)*x_0
-x_1 + z_12^2*x_2
(z_12^2 - 1)*x_1
(z_12^2 - 1)*x_3
(z_12^2 - 1)*x_4 + (-z_12^2 + 1)*x_5
-z_12^2*x_5
x_3 + x_6
and inverse
-z_12^2*x_0
-z_12^2*x_2
(-z_12^2 + 1)*x_1 - x_2
-z_12^2*x_3
-z_12^2*x_4 + (z_12^2 - 1)*x_5
(z_12^2 - 1)*x_5
z_12^2*x_3 + x_6
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition, ReesAlgebra, TangentCone
i1 : loadPackage("Cyclotomic")
o1 = Cyclotomic
o1 : Package
i2 : K=cyclotomicField(12);
i3 : z_12=K_0;
i4 : z_6=z_12^2;
i5 : z_3=z_12^4;
i6 : S=K[x_0..x_6];
i7 : c_6=z_12^3+1;
i8 : I=ideal({
x_0^2 + x_3*x_4 - z_12^2*x_3*x_5 - z_12^2*x_5*x_6,
x_1^2 +x_3*x_4 + z_12^2*x_3*x_5 + 2*x_4*x_6 - z_12^2*x_5*x_6,
x_1*x_2 + (-2*z_12^2 + 1)*x_3*x_4 + z_12^2*x_3*x_5 - 2*z_12^2*x_4*x_6 + z_12^2*x_5*x_6,
x_2^2 + -x_3*x_4 + (-z_12^2 + 2)*x_3*x_5 + (2*z_12^2 - 2)*x_4*x_6 + (-z_12^2 + 2)*x_5*x_6,
x_1*x_3 - z_12^2*x_2*x_6 + c_6*(z_12*x_4^2 + (z_12^3 - 2*z_12)*x_4*x_5 + z_12*x_5^2),
x_1*x_4 + (-z_12^2 + 1)*x_2*x_5 + c_6*(-x_3*x_6 - x_6^2),
x_1*x_5 - x_2*x_5 + c_6*(x_3^2 + (-z_12^2 + 2)*x_3*x_6),
x_1*x_6 + x_2*x_6 + c_6*(-z_12*x_4^2 + z_12*x_4*x_5 - z_12*x_5^2),
x_2*x_3 - z_12^2*x_2*x_6 + c_6*(z_12*x_4^2),
x_2*x_4 + (-z_12^2 - 1)*x_2*x_5 + c_6*(z_12^2*x_3^2 + 2*z_12^2*x_3*x_6 + z_12^2*x_6^2)
});
o8 : Ideal of S
i9 : betti res I
0 1 2 3 4 5
o9 = total: 1 10 25 25 10 1
0: 1 . . . . .
1: . 10 16 9 . .
2: . . 9 16 10 .
3: . . . . . 1
o9 : BettiTally