Equations of a 1-parameter family of genus 4 Riemann surfaces with automorphism group (18,3)

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 1-parameter family of genus 7 Riemann surfaces with automorphism group (18,3) in the GAP library of small groups. The quotient of any surface in this family has genus zero, and the quotient morphism is branched over four points with ramification indices (2,2,3,3).

We use Magma to compute equations of one member of this family, and give a conjectural description of this family.

Obtaining candidate polynomials in Magma

We use some Magma code developed by David Swinarski during a visit to the University of Sydney in June/July 2011. Here is the file autcv10e.txt used below.

Magma V2.21-7     Mon Apr 25 2016 20:09:42 on Davids-MacBook-Pro-2 [Seed = 
2294378005]

+-------------------------------------------------------------------+
|       This copy of Magma has been made available through a        |
|                   generous initiative of the                      |
|                                                                   |
|                         Simons Foundation                         |
|                                                                   |
| covering U.S. Colleges, Universities, Nonprofit Research entities,|
|               and their students, faculty, and staff              |
+-------------------------------------------------------------------+

Type ? for help.  Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> load "autcv10e.txt";
Loading "autcv10e.txt"
> G:=SmallGroup(18,3);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,4,[2,2,3,3]);
Set seed to 0.


Character Table of Group G
--------------------------


-------------------------------------------
Class |   1  2    3    4  5   6   7   8   9
Size  |   1  3    1    1  2   2   2   3   3
Order |   1  2    3    3  3   3   3   6   6
-------------------------------------------
p  =  2   1  1    4    3  5   7   6   3   4
p  =  3   1  2    1    1  1   1   1   2   2
-------------------------------------------
X.1   +   1  1    1    1  1   1   1   1   1
X.2   +   1 -1    1    1  1   1   1  -1  -1
X.3   0   1 -1 -1-J    J  1-1-J   J  -J 1+J
X.4   0   1 -1    J -1-J  1   J-1-J 1+J  -J
X.5   0   1  1    J -1-J  1   J-1-J-1-J   J
X.6   0   1  1 -1-J    J  1-1-J   J   J-1-J
X.7   +   2  0    2    2 -1  -1  -1   0   0
X.8   0   2  0  2*J-2-2*J -1  -J 1+J   0   0
X.9   0   2  0-2-2*J  2*J -1 1+J  -J   0   0


Explanation of Character Value Symbols
--------------------------------------

J = RootOfUnity(3)


Conjugacy Classes of group G
----------------------------
[1]     Order 1       Length 1      
        Rep Id(G)

[2]     Order 2       Length 3      
        Rep G.1

[3]     Order 3       Length 1      
        Rep G.2^2

[4]     Order 3       Length 1      
        Rep G.2

[5]     Order 3       Length 2      
        Rep G.3

[6]     Order 3       Length 2      
        Rep G.2^2 * G.3

[7]     Order 3       Length 2      
        Rep G.2 * G.3

[8]     Order 6       Length 3      
        Rep G.1 * G.2

[9]     Order 6       Length 3      
        Rep G.1 * G.2^2


Surface kernel generators:  [ G.1 * G.3^2, G.1, G.2, G.2^2 * G.3^2 ]
Is hyperelliptic?  false
Is cyclic trigonal?  true
Multiplicities of irreducibles in relevant G-modules:
I_1      =[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1      =[ 0, 0, 1, 1, 0, 0, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 1, 1, 0, 0, 0, 0, 1 ]
I_2      =[ 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
S_2      =[ 1, 0, 0, 0, 2, 1, 1, 2, 0 ]
H^0(C,2K)=[ 1, 0, 0, 0, 1, 1, 1, 2, 0 ]
I_3      =[ 0, 2, 1, 0, 0, 0, 1, 0, 0 ]
S_3      =[ 1, 4, 2, 1, 0, 0, 3, 1, 2 ]
H^0(C,3K)=[ 1, 2, 1, 1, 0, 0, 2, 1, 2 ]
I2timesS1=[ 0, 1, 1, 0, 0, 0, 1, 0, 0 ]
Is clearly not generated by quadrics? true
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 18 and degree 6
[
    [ -1   0   0   0]
    [  0  -1   0   0]
    [  0   0   1   0]
    [  0   0 z^3  -1],

    [z^3 - 1       0       0       0]
    [      0    -z^3       0       0]
    [      0       0 z^3 - 1       0]
    [      0       0       0 z^3 - 1],

    [      1       0       0       0]
    [      0       1       0       0]
    [      0       0       0 z^3 - 1]
    [      0       0     z^3      -1]
]
Matrix Surface Kernel Generators:
[
    [      -1        0        0        0]
    [       0       -1        0        0]
    [       0        0       -1 -z^3 + 1]
    [       0        0        0        1],

    [ -1   0   0   0]
    [  0  -1   0   0]
    [  0   0   1   0]
    [  0   0 z^3  -1],

    [z^3 - 1       0       0       0]
    [      0    -z^3       0       0]
    [      0       0 z^3 - 1       0]
    [      0       0       0 z^3 - 1],

    [   -z^3       0       0       0]
    [      0 z^3 - 1       0       0]
    [      0       0     z^3      -1]
    [      0       0 z^3 - 1       0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 5
Dimension 2
Multiplicity 2
[
    x_0^2,
    x_2^2 + z^3*x_2*x_3 + (z^3 - 1)*x_3^2
]
Finding cubics:
I3 contains a 2-dimensional subspace of CharacterRow 2
Dimension 4
Multiplicity 4
[
    x_0^3,
    x_0*x_2^2 + z^3*x_0*x_2*x_3 + (z^3 - 1)*x_0*x_3^2,
    x_1^3,
    x_2^2*x_3 + z^3*x_2*x_3^2
]
I3 contains a 1-dimensional subspace of CharacterRow 3
Dimension 2
Multiplicity 2
[
    x_0^2*x_1,
    x_1*x_2^2 + z^3*x_1*x_2*x_3 + (z^3 - 1)*x_1*x_3^2
]
I3 contains a 2-dimensional subspace of CharacterRow 7
Dimension 6
Multiplicity 3
[
    x_0^2*x_2,
    x_0^2*x_3,
    x_0*x_2^2 + (-z^3 + 1)*x_0*x_3^2,
    x_0*x_2*x_3 + 1/2*z^3*x_0*x_3^2,
    x_2^3 + x_3^3,
    x_2^2*x_3 + z^3*x_2*x_3^2 + (z^3 - 1)*x_3^3
]

From the output, we see that this curve is cyclic trigonal. The ideal contains a quadric of the form \[ c_1(x_0^2) +c_2(x_2^2 + \zeta_6 x_2*x_3 + \zeta_3 x_3^2) \] We may assume that the extra cubic generator is of the form \[ c_3(x_0^3)+ c_4(x_1^3) +c_5(x_2^2*x_3 + \zeta_6 x_2*x_3^2) \] Assume that \(c_1\) and \(c_2\) are nonzero. Then after scaling \(x_0\) and dividing, we may assume that \(c_1=c_2=1\). Similarly, after scaling \(x_1\) and dividing, we may assume that \(c_3=c_4=1\).

Checking the equations in `Magma`

We check that for two different values of \(c_5\) these equations give a smooth genus 4 curve with the desired automorphisms. This implies that a general member of the pencil is a smooth curve with the correct automorphisms. However, I did not check that the two curves studied below are not isomorphic to each other; it is possible that we have described a point in \( \mathcal{M}_{g}\) rather than a curve in \( \mathcal{M}_g\). First we study the value \(c_5 =1\):

> K<z_6>:=CyclotomicField(6);
> z_3:=z_6^2;
> P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
> c_5:=1;
> X:=Scheme(P3,[
> x_0^2 + x_2^2 + z_6*x_2*x_3 + z_3*x_3^2,
> x_0^3 + x_1^3 + c_5*(x_2^2*x_3 + z_6*x_2*x_3^2)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
6*$.1 - 3
2
> A:=Matrix([
> [-1,0,0,0],
> [0,-1,0,0],
> [0,0,-1,z_6^-1],
> [0,0,0,1]
> ]);
> B:=Matrix([
> [-1,0,0,0],
> [0,-1,0,0],
> [0,0,1,0],
> [0,0,z_6,-1]
> ]);
> C:=Matrix([
> [z_3,0,0,0],
> [0,-z_6,0,0],
> [0,0,z_3,0],
> [0,0,0,z_3]
> ]);
> Order(A);
2
> Order(B);
2
> Order(C);
3
> Order( (A*B*C)^-1);
3
> GL4K:=GeneralLinearGroup(4,K);
> IdentifyGroup(sub<GL4K | A,B,C>);
<18, 3>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
-x_2
(-z_6 + 1)*x_2 + x_3
and inverse
-x_0
-x_1
-x_2
(-z_6 + 1)*x_2 + x_3
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
x_2 + z_6*x_3
-x_3
and inverse
-x_0
-x_1
x_2 + z_6*x_3
-x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
(z_6 - 1)*x_0
-z_6*x_1
(z_6 - 1)*x_2
(z_6 - 1)*x_3
and inverse
-z_6*x_0
(z_6 - 1)*x_1
-z_6*x_2
-z_6*x_3

Next we study the value \(c_5 =17-\zeta_6^5\):

> c_5:=17-z_6^-1;
> X:=Scheme(P3,[
> x_0^2 + x_2^2 + z_6*x_2*x_3 + z_3*x_3^2,
> x_0^3 + x_1^3 + c_5*(x_2^2*x_3 + z_6*x_2*x_3^2)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
6*$.1 - 3
2
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
-x_2
(-z_6 + 1)*x_2 + x_3
and inverse
-x_0
-x_1
-x_2
(-z_6 + 1)*x_2 + x_3
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
x_2 + z_6*x_3
-x_3
and inverse
-x_0
-x_1
x_2 + z_6*x_3
-x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
(z_6 - 1)*x_0
-z_6*x_1
(z_6 - 1)*x_2
(z_6 - 1)*x_3
and inverse
-z_6*x_0
(z_6 - 1)*x_1
-z_6*x_2
-z_6*x_3

Cyclic trigonal approach

This family contains the genus 4 Riemann surface with automorphism group (36,12). We use this to obtain trigonal equations for the surfaces in this family. When we analyzed the Riemann surface with automorphism group (36,12), we found matrix generators of a subgroup of \(GL(4,\mathbb{C})\) isomorphic to (36,12). First we look for a subgroup of this matrix group isomorphic to (18,3).

> A:=Matrix([
> [-1,0,0,0],
> [0,0,0,-z_6],
> [0,0,-1,0],
> [0,z_6-1,0,0]
> ]);
> B:=Matrix([
> [z_3,0,0,0],
> [0,0,0,-1],
> [0,0,z_6,0],
> [0,-z_3,0,0]
> ]);
> G:=sub<GL4K | A,B>;
> SCL:=SubgroupClasses(G);
> SCL;
Conjugacy classes of subgroups
------------------------------

[ 1]    Order 1            Length 1
        MatrixGroup(4, K) of order 1
[ 2]    Order 2            Length 1
        MatrixGroup(4, K) of order 2
        Generators:
            [-1  0  0  0]
            [ 0 -1  0  0]
            [ 0  0  1  0]
            [ 0  0  0 -1]
[ 3]    Order 2            Length 3
        MatrixGroup(4, K) of order 2
        Generators:
            [     -1       0       0       0]
            [      0       0       0    -z_6]
            [      0       0      -1       0]
            [      0 z_6 - 1       0       0]
[ 4]    Order 2            Length 3
        MatrixGroup(4, K) of order 2
        Generators:
            [       1        0        0        0]
            [       0        0        0 -z_6 + 1]
            [       0        0       -1        0]
            [       0      z_6        0        0]
[ 5]    Order 3            Length 1
        MatrixGroup(4, K) of order 3
        Generators:
            [      1       0       0       0]
            [      0    -z_6       0       0]
            [      0       0       1       0]
            [      0       0       0 z_6 - 1]
[ 6]    Order 3            Length 1
        MatrixGroup(4, K) of order 3
        Generators:
            [   -z_6       0       0       0]
            [      0 z_6 - 1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0 z_6 - 1]
[ 7]    Order 3            Length 2
        MatrixGroup(4, K) of order 3
        Generators:
            [   -z_6       0       0       0]
            [      0       1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0    -z_6]
[ 8]    Order 4            Length 3
        MatrixGroup(4, K) of order 2^2
        Generators:
            [       1        0        0        0]
            [       0        0        0 -z_6 + 1]
            [       0        0       -1        0]
            [       0      z_6        0        0]

            [     -1       0       0       0]
            [      0       0       0 z_6 - 1]
            [      0       0      -1       0]
            [      0    -z_6       0       0]
[ 9]    Order 6            Length 1
        MatrixGroup(4, K) of order 2 * 3
        Generators:
            [-1  0  0  0]
            [ 0 -1  0  0]
            [ 0  0  1  0]
            [ 0  0  0 -1]

            [   -z_6       0       0       0]
            [      0 z_6 - 1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0 z_6 - 1]
[10]    Order 6            Length 1
        MatrixGroup(4, K) of order 2 * 3
        Generators:
            [      -1        0        0        0]
            [       0      z_6        0        0]
            [       0        0        1        0]
            [       0        0        0 -z_6 + 1]

            [      1       0       0       0]
            [      0    -z_6       0       0]
            [      0       0       1       0]
            [      0       0       0 z_6 - 1]
[11]    Order 6            Length 1
        MatrixGroup(4, K) of order 2 * 3
        Generators:
            [     -1       0       0       0]
            [      0       0       0    -z_6]
            [      0       0      -1       0]
            [      0 z_6 - 1       0       0]

            [      1       0       0       0]
            [      0    -z_6       0       0]
            [      0       0       1       0]
            [      0       0       0 z_6 - 1]
[12]    Order 6            Length 1
        MatrixGroup(4, K) of order 2 * 3
        Generators:
            [       1        0        0        0]
            [       0        0        0 -z_6 + 1]
            [       0        0       -1        0]
            [       0      z_6        0        0]

            [      1       0       0       0]
            [      0    -z_6       0       0]
            [      0       0       1       0]
            [      0       0       0 z_6 - 1]
[13]    Order 6            Length 2
        MatrixGroup(4, K) of order 2 * 3
        Generators:
            [-1  0  0  0]
            [ 0 -1  0  0]
            [ 0  0  1  0]
            [ 0  0  0 -1]

            [   -z_6       0       0       0]
            [      0       1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0    -z_6]
[14]    Order 6            Length 3
        MatrixGroup(4, K) of order 2 * 3
        Generators:
            [       1        0        0        0]
            [       0        0        0 -z_6 + 1]
            [       0        0       -1        0]
            [       0      z_6        0        0]

            [   -z_6       0       0       0]
            [      0 z_6 - 1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0 z_6 - 1]
[15]    Order 6            Length 3
        MatrixGroup(4, K) of order 2 * 3
        Generators:
            [     -1       0       0       0]
            [      0       0       0    -z_6]
            [      0       0      -1       0]
            [      0 z_6 - 1       0       0]

            [   -z_6       0       0       0]
            [      0 z_6 - 1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0 z_6 - 1]
[16]    Order 9            Length 1
        MatrixGroup(4, K) of order 3^2
        Generators:
            [   -z_6       0       0       0]
            [      0 z_6 - 1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0 z_6 - 1]

            [      1       0       0       0]
            [      0    -z_6       0       0]
            [      0       0       1       0]
            [      0       0       0 z_6 - 1]
[17]    Order 12           Length 1
        MatrixGroup(4, K) of order 2^2 * 3
        Generators:
            [       1        0        0        0]
            [       0        0        0 -z_6 + 1]
            [       0        0       -1        0]
            [       0      z_6        0        0]

            [     -1       0       0       0]
            [      0       0       0    -z_6]
            [      0       0      -1       0]
            [      0 z_6 - 1       0       0]

            [      1       0       0       0]
            [      0    -z_6       0       0]
            [      0       0       1       0]
            [      0       0       0 z_6 - 1]
[18]    Order 12           Length 3
        MatrixGroup(4, K) of order 2^2 * 3
        Generators:
            [       1        0        0        0]
            [       0        0        0 -z_6 + 1]
            [       0        0       -1        0]
            [       0      z_6        0        0]

            [     -1       0       0       0]
            [      0       0       0 z_6 - 1]
            [      0       0      -1       0]
            [      0    -z_6       0       0]

            [   -z_6       0       0       0]
            [      0 z_6 - 1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0 z_6 - 1]
[19]    Order 18           Length 1
        MatrixGroup(4, K) of order 2 * 3^2
        Generators:
            [     -1       0       0       0]
            [      0       0       0    -z_6]
            [      0       0      -1       0]
            [      0 z_6 - 1       0       0]

            [   -z_6       0       0       0]
            [      0 z_6 - 1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0 z_6 - 1]

            [      1       0       0       0]
            [      0    -z_6       0       0]
            [      0       0       1       0]
            [      0       0       0 z_6 - 1]
[20]    Order 18           Length 1
        MatrixGroup(4, K) of order 2 * 3^2
        Generators:
            [      -1        0        0        0]
            [       0      z_6        0        0]
            [       0        0        1        0]
            [       0        0        0 -z_6 + 1]

            [   -z_6       0       0       0]
            [      0 z_6 - 1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0 z_6 - 1]

            [      1       0       0       0]
            [      0    -z_6       0       0]
            [      0       0       1       0]
            [      0       0       0 z_6 - 1]
[21]    Order 18           Length 1
        MatrixGroup(4, K) of order 2 * 3^2
        Generators:
            [       1        0        0        0]
            [       0        0        0 -z_6 + 1]
            [       0        0       -1        0]
            [       0      z_6        0        0]

            [   -z_6       0       0       0]
            [      0 z_6 - 1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0 z_6 - 1]

            [      1       0       0       0]
            [      0    -z_6       0       0]
            [      0       0       1       0]
            [      0       0       0 z_6 - 1]
[22]    Order 36           Length 1
        MatrixGroup(4, K) of order 2^2 * 3^2
        Generators:
            [       1        0        0        0]
            [       0        0        0 -z_6 + 1]
            [       0        0       -1        0]
            [       0      z_6        0        0]

            [     -1       0       0       0]
            [      0       0       0    -z_6]
            [      0       0      -1       0]
            [      0 z_6 - 1       0       0]

            [   -z_6       0       0       0]
            [      0 z_6 - 1       0       0]
            [      0       0 z_6 - 1       0]
            [      0       0       0 z_6 - 1]

            [      1       0       0       0]
            [      0    -z_6       0       0]
            [      0       0       1       0]
            [      0       0       0 z_6 - 1]
> IdentifyGroup(SCL[19]`subgroup);
<18, 3>
> IdentifyGroup(SCL[20]`subgroup);
<18, 5>
> IdentifyGroup(SCL[21]`subgroup);
<18, 3>

The output shows two subgroups of (36,12) isomorphic to (18,3). We find matrix surface kernel generators for one of these subgroups and proceed with the analysis:

> H:=SCL[19]`subgroup;             
> ASKG:=AllSurfaceKernelGenerators(H,[2,2,3,3]);
> MatrixGens,MatrixSKG,Q,C:=RunGivenSKMatrixGenerators(18,4,ASKG[1]);
Set seed to 0.


Character Table of Group G
--------------------------


-------------------------------------------
Class |   1  2    3    4  5   6   7   8   9
Size  |   1  3    1    1  2   2   2   3   3
Order |   1  2    3    3  3   3   3   6   6
-------------------------------------------
p  =  2   1  1    4    3  5   7   6   3   4
p  =  3   1  2    1    1  1   1   1   2   2
-------------------------------------------
X.1   +   1  1    1    1  1   1   1   1   1
X.2   +   1 -1    1    1  1   1   1  -1  -1
X.3   0   1 -1 -1-J    J  1-1-J   J  -J 1+J
X.4   0   1 -1    J -1-J  1   J-1-J 1+J  -J
X.5   0   1  1    J -1-J  1   J-1-J-1-J   J
X.6   0   1  1 -1-J    J  1-1-J   J   J-1-J
X.7   +   2  0    2    2 -1  -1  -1   0   0
X.8   0   2  0  2*J-2-2*J -1  -J 1+J   0   0
X.9   0   2  0-2-2*J  2*J -1 1+J  -J   0   0


Explanation of Character Value Symbols
--------------------------------------

J = RootOfUnity(3)


Conjugacy Classes of group G
----------------------------
[1]     Order 1       Length 1      
        Rep [1 0 0 0]
        [0 1 0 0]
        [0 0 1 0]
        [0 0 0 1]

[2]     Order 2       Length 3      
        Rep [     -1       0       0       0]
        [      0       0       0 z^3 - 1]
        [      0       0      -1       0]
        [      0    -z^3       0       0]

[3]     Order 3       Length 1      
        Rep [   -z^3       0       0       0]
        [      0 z^3 - 1       0       0]
        [      0       0 z^3 - 1       0]
        [      0       0       0 z^3 - 1]

[4]     Order 3       Length 1      
        Rep [z^3 - 1       0       0       0]
        [      0    -z^3       0       0]
        [      0       0    -z^3       0]
        [      0       0       0    -z^3]

[5]     Order 3       Length 2      
        Rep [      1       0       0       0]
        [      0    -z^3       0       0]
        [      0       0       1       0]
        [      0       0       0 z^3 - 1]

[6]     Order 3       Length 2      
        Rep [   -z^3       0       0       0]
        [      0    -z^3       0       0]
        [      0       0 z^3 - 1       0]
        [      0       0       0       1]

[7]     Order 3       Length 2      
        Rep [z^3 - 1       0       0       0]
        [      0 z^3 - 1       0       0]
        [      0       0    -z^3       0]
        [      0       0       0       1]

[8]     Order 6       Length 3      
        Rep [-z^3 + 1        0        0        0]
        [       0        0        0        1]
        [       0        0      z^3        0]
        [       0  z^3 - 1        0        0]

[9]     Order 6       Length 3      
        Rep [     z^3        0        0        0]
        [       0        0        0     -z^3]
        [       0        0 -z^3 + 1        0]
        [       0        1        0        0]


Is hyperelliptic?  false
Is cyclic trigonal?  true
Multiplicities of irreducibles in relevant G-modules:
I_1      =[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1      =[ 0, 0, 1, 1, 0, 0, 0, 1, 0 ]
H^0(C,K) =[ 0, 0, 1, 1, 0, 0, 0, 1, 0 ]
I_2      =[ 0, 0, 0, 0, 0, 1, 0, 0, 0 ]
S_2      =[ 1, 0, 0, 0, 1, 2, 1, 0, 2 ]
H^0(C,2K)=[ 1, 0, 0, 0, 1, 1, 1, 0, 2 ]
I_3      =[ 0, 2, 0, 1, 0, 0, 1, 0, 0 ]
S_3      =[ 1, 4, 1, 2, 0, 0, 3, 2, 1 ]
H^0(C,3K)=[ 1, 2, 1, 1, 0, 0, 2, 2, 1 ]
I2timesS1=[ 0, 1, 0, 1, 0, 0, 1, 0, 0 ]
Is clearly not generated by quadrics? true
Matrix Surface Kernel Generators:
Field K Cyclotomic Field of order 18 and degree 6
[
    [     -1       0       0       0]
    [      0       0       0 z^3 - 1]
    [      0       0      -1       0]
    [      0    -z^3       0       0],

    [-1  0  0  0]
    [ 0  0  0  1]
    [ 0  0 -1  0]
    [ 0  1  0  0],

    [   -z^3       0       0       0]
    [      0 z^3 - 1       0       0]
    [      0       0 z^3 - 1       0]
    [      0       0       0 z^3 - 1],

    [z^3 - 1       0       0       0]
    [      0 z^3 - 1       0       0]
    [      0       0    -z^3       0]
    [      0       0       0       1]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 6
Dimension 2
Multiplicity 2
[
    x_1*x_3,
    x_2^2
]
Finding cubics:
I3 contains a 2-dimensional subspace of CharacterRow 2
Dimension 4
Multiplicity 4
[
    x_0^3,
    x_1^3 - x_3^3,
    x_1*x_2*x_3,
    x_2^3
]
I3 contains a 1-dimensional subspace of CharacterRow 4
Dimension 2
Multiplicity 2
[
    x_0*x_1*x_3,
    x_0*x_2^2
]
I3 contains a 2-dimensional subspace of CharacterRow 7
Dimension 6
Multiplicity 3
[
    x_1^2*x_2,
    x_1^2*x_3,
    x_1*x_2^2,
    x_1*x_3^2,
    x_2^2*x_3,
    x_2*x_3^2
]

This shows that the canonical ideal is \[ x_1x_3-x_2^2, x_0^3 + x_1^3-x_3^3+c x_2^2 \] Comparing this to the basis of differentials \[ \{y \frac{dx}{y^2}, \frac{dx}{y^2}, x \frac{dx}{y^2}, x^2 \frac{dx}{y^2} \} \] we see that this cubic encodes \( y^3 = (x^3+t)(x^3-t^{-1}) \)

Equations of a 1-parameter family of genus 4 Riemann surfaces with automorphism group (18,3)

Obtaining candidate polynomials in Magma

Checking the equations in Magma

Cyclic trigonal approach

Checking the equations in `Magma`