Fordham University

Finding equations for two genus 7 Riemann surfaces with 54 automorphisms

Magaard, Shaska, Shpectorov, and Völklein give tables of 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.

Their tables list two genus 7 Riemann surfaces with automorphism group (54,6) in the GAP library of small groups. The quotients of each of these surfaces by its automorphism group have genus zero, and the quotient morphisms are branched over three points with ramification indices (2,6,9).

We use Magma, GAP and Macaulay2 to compute equations of these two Riemann surfaces from Breuer's data.

Most of the discussion below focuses on calculations to obtain equations of one of these two surfaces. Then, at the end, we show that equations of the second surface may be obtained as the complex conjugates of the equations of the first.

Two sets of surface kernel generators

I downloaded Breuer's data from Paulhus's website . In the file "grpmongap07" (located in the section GAP: full data to genus 40 on Paulhus's website), lines 1011-1012, Breuer lists the following two sets of surface kernel generators: 
[ 54, 6 ][ 0, 2, 6, 9 ][ 2, 6, 10 ][ f1, f1*f2*f3^2*f4, f2^2*f3*f4^2 ]
[ 54, 6 ][ 0, 2, 6, 9 ][ 2, 9, 8 ][ f1, f1*f2^2*f3^2*f4^2, f2*f3 ]
Here the generators f1 through f4 are with respect to the polycyclic group structure in which the group (54,6) is stored in GAP's library. We begin by checking that Magma and GAP use the same description of the group (54,6). We show that the GAP relations are satisfied by the Magma group, and the Magma relations are satisfied by the GAP group. First we obtain the GAP relations: 
GAP 4.8.2, 20-Feb-2016, build of 2016-03-12 10:04:43 (EST)
gap> p:=FamilyPcgs( SmallGroup(54,6));  
Pcgs([ f1, f2, f3, f4 ])
gap> iso:=IsomorphismFpGroupByPcgs( p, "G.");
[ f1, f2, f3, f4 ] -> [ G.1, G.2, G.3, G.4 ]
gap> RelatorsOfFpGroup(Image(iso));
[ G.1^2, G.2^-1*G.1^-1*G.2*G.1, G.3^-1*G.1^-1*G.3*G.1*G.4^-1*G.3^-1, 
  G.4^-1*G.1^-1*G.4*G.1*G.4^-1, G.2^3, G.3^-1*G.2^-1*G.3*G.2*G.4^-1, 
  G.4^-1*G.2^-1*G.4*G.2, G.3^3*G.4^-2, G.4^-1*G.3^-1*G.4*G.3, G.4^3 ]
We check that these are satisfied by the Magma group: 
Magma V2.21-10    Sat Mar 12 2016 10:28:09 on ace-math01 [Seed = 3228738308]
Type ? for help.  Type -D to quit.
> G:=SmallGroup(54,6);
> [ G.1^2, G.2^-1*G.1^-1*G.2*G.1, G.3^-1*G.1^-1*G.3*G.1*G.4^-1*G.3^-1, 
>   G.4^-1*G.1^-1*G.4*G.1*G.4^-1, G.2^3, G.3^-1*G.2^-1*G.3*G.2*G.4^-1, 
>   G.4^-1*G.2^-1*G.4*G.2, G.3^3*G.4^-2, G.4^-1*G.3^-1*G.4*G.3, G.4^3 ];
[ Id(G), Id(G), Id(G), Id(G), Id(G), Id(G), Id(G), Id(G), Id(G), Id(G) ]
Next we obtain the Magma relations: 
> G;
GrpPC : G of order 54 = 2 * 3^3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^3 = Id(G), 
    G.3^3 = G.4^2, 
    G.4^3 = Id(G), 
    G.3^G.1 = G.3^2 * G.4, 
    G.3^G.2 = G.3 * G.4, 
    G.4^G.1 = G.4^2
Recall that in Magma, g ^ h is notation for h^(-1)*g*h. We check that these relations are satisfied by the GAP group: 
gap> p[1]^2;
<identity> of ...
gap> p[2]^3;
<identity> of ...
gap> p[3]^3*(p[4]^2)^(-1);
<identity> of ...
gap> p[4]^3;
<identity> of ...
gap> p[1]^(-1)*p[3]*p[1]*(p[3]^2 * p[4])^(-1);
<identity> of ...
gap> p[2]^(-1)*p[3]*p[2]*(p[3] * p[4])^(-1); 
<identity> of ...
gap> p[1]^(-1)*p[4]*p[1]*(p[4]^2)^(-1);
<identity> of ...
This shows that Magma and GAP use the same polycyclic presentation of the group (54,6).

Obtaining candidate polynomials for the first curve in Magma and GAP

We analyze Breuer's first set of surface kernel generators. 
> load "autcv10c.txt";  
Loading "autcv10c.txt"
> G:=SmallGroup(54,6);
> RunGivenSKG(G,7,[G.1, G.1*G.2*G.3^2*G.4, G.2^2*G.3*G.4^2]);
Set seed to 0.


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


--------------------------------------------------
Class |   1  2  3      4      5   6   7   8  9  10
Size  |   1  9  2      3      3   9   9   6  6   6
Order |   1  2  3      3      3   6   6   9  9   9
--------------------------------------------------
p  =  2   1  1  3      5      4   4   5  10  9   8
p  =  3   1  2  1      1      1   2   2   3  3   3
--------------------------------------------------
X.1   +   1  1  1      1      1   1   1   1  1   1
X.2   +   1 -1  1      1      1  -1  -1   1  1   1
X.3   0   1 -1  1      J   -1-J 1+J  -J-1-J  1   J
X.4   0   1  1  1   -1-J      J   J-1-J   J  1-1-J
X.5   0   1  1  1      J   -1-J-1-J   J-1-J  1   J
X.6   0   1 -1  1   -1-J      J  -J 1+J   J  1-1-J
X.7   +   2  0  2      2      2   0   0  -1 -1  -1
X.8   0   2  0  2    2*J -2-2*J   0   0 1+J -1  -J
X.9   0   2  0  2 -2-2*J    2*J   0   0  -J -1 1+J
X.10  +   6  0 -3      0      0   0   0   0  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 9      
        Rep G.1

[3]     Order 3       Length 2      
        Rep G.4

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

[5]     Order 3       Length 3      
        Rep G.2

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

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

[8]     Order 9       Length 6      
        Rep G.2 * G.3 * G.4

[9]     Order 9       Length 6      
        Rep G.3

[10]    Order 9       Length 6      
        Rep G.2^2 * G.3 * G.4^2


Surface kernel generators:  [ G.1, G.1 * G.2 * G.3^2 * G.4, G.2^2 * G.3 * G.4^2 
]
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 ]
S_1      =[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 ]
I_2      =[ 1, 0, 0, 1, 0, 0, 0, 1, 0, 1 ]
S_2      =[ 1, 0, 0, 2, 1, 0, 1, 1, 1, 3 ]
H^0(C,2K)=[ 0, 0, 0, 1, 1, 0, 1, 0, 1, 2 ]
I_3      =[ 1, 2, 1, 1, 0, 1, 2, 2, 2, 6 ]
S_3      =[ 1, 3, 2, 1, 1, 2, 4, 3, 3, 9 ]
H^0(C,3K)=[ 0, 1, 1, 0, 1, 1, 2, 1, 1, 3 ]
I2timesS1=[ 1, 2, 2, 1, 1, 1, 2, 2, 3, 8 ]
Is clearly not generated by quadrics? false
No subgroup found

RunGivenSKG(
    G: GrpPC : G,
    genus: 7,
    SKG: [ G.1, G.1 * G.2 * G.3^2 * G.4, G.2^2 * G.3 * G.4^2 ]
)
FindMatrixGenerators(
    G: GrpPC : G,
    genus: 7,
    T:   Character Table of Group G --------------------------   --...,
    CCL: Conjugacy Classes of group G ---------------------------- [1...,
    M: [ G.1, G.1 * G.2 * G.3^2 * G.4, G.2^2 * G.3 * G.4^2 ]
)
In file "autcv10c.txt", line 220, column 28:
>>       ags:=ActionGenerators(GModule(T[i]));
                              ^
Runtime error in 'ActionGenerators': Bad argument types
Argument types given: BoolElt

The error "No subgroup found" tells us that Magma has an internal error when finding the matrix generators of representations with character \( \chi_{3}\) and \( \chi_{10}\). Fortunately, we can obtain these matrix generators from GAP. 


gap> IrreducibleRepresentations(G);
[ Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ E(3) ] ], [ [ 1 ] ], [ [ 1 ] ] 
     ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ E(3) ] ], [ [ 1 ] ], 
      [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ E(3)^2 ] ], [ [ 1 ] ], 
      [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ E(3)^2 ] ], [ [ 1 ] ], 
      [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4 ]) -> 
    [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ], 
      [ [ E(3), 0 ], [ 0, E(3)^2 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4 ]) -> 
    [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(3), 0 ], [ 0, E(3) ] ], 
      [ [ E(3), 0 ], [ 0, E(3)^2 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4 ]) -> 
    [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(3)^2, 0 ], [ 0, E(3)^2 ] ], 
      [ [ E(3), 0 ], [ 0, E(3)^2 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4 ]) -> 
    [ [ [ 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1, 0 ], 
          [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0 ] ], 
      [ [ 0, 0, 1, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], 
          [ 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1, 0 ] ], 
      [ [ E(9)^2, 0, 0, 0, 0, 0 ], [ 0, E(9)^5, 0, 0, 0, 0 ], 
          [ 0, 0, -E(9)^2-E(9)^5, 0, 0, 0 ], [ 0, 0, 0, E(9)^4, 0, 0 ], 
          [ 0, 0, 0, 0, -E(9)^4-E(9)^7, 0 ], [ 0, 0, 0, 0, 0, E(9)^7 ] ], 
      [ [ E(3), 0, 0, 0, 0, 0 ], [ 0, E(3), 0, 0, 0, 0 ], 
          [ 0, 0, E(3), 0, 0, 0 ], [ 0, 0, 0, E(3)^2, 0, 0 ], 
          [ 0, 0, 0, 0, E(3)^2, 0 ], [ 0, 0, 0, 0, 0, E(3)^2 ] ] ] ]

We load these matrices into Magma, compute the characters of the representations they give, and compare them to the order of the characters in the Magma character table. 

> K:=CyclotomicField(9);
> rho:=function(G,K,L)
function>     n:=NumberOfRows(Matrix(L[1]));
function>     GLnK:=GeneralLinearGroup(n,K);
function>     L:=[GLnK!Matrix(L[i]): i in [1..#L]];
function>     return hom< G -> GLnK | L>;
function> end function;
> char:=function(CCLR,f)
function>     return [Trace(f(CCLR[i])) : i in [1..#CCLR]];
function> end function;
> LookupCharacter:=function(T,chi)
function>     for i:=1 to #T do
function|for>         if T[i] eq chi then
function|for|if>             return i;
function|for|if>         end if;
function|for>     end for;
function> end function;
> L:=[
> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ 1 ] ], [ [ z^3 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ -1 ] ], [ [ z^3 ] ], [ [ 1 ] ],[ [ 1 ] ] ],
> [ [ [ 1 ] ], [ [ (z^3)^2 ] ], [ [ 1 ] ],[ [ 1 ] ] ],
> [ [ [ -1 ] ], [ [(z^3)^2 ] ], [ [ 1 ] ],[ [ 1 ] ] ],
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ],[ [ z^3, 0 ], [ 0, (z^3)^2]\
], [ [ 1, 0 ], [ 0, 1 ] ] ],
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [z^3, 0 ], [ 0, z^3 ] ],[ [ z^3, 0 ], [ 0, (z^3)\
^2 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ],
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ (z^3)^2, 0 ], [ 0, (z^3)^2 ] ],[ [ z^3, 0 ], [\
0, (z^3)^2 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ],
> [ [ [ 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1, 0 ],[ 0, 1,\
0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0 ] ],[ [ 0, 0, 1, 0, 0,0\
], [ 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ],[ 0, 0, 0, 0, 0, 1 ], [ 0, 0,0,1\
, 0, 0 ], [ 0, 0, 0, 0, 1, 0 ] ],[ [ (z)^2, 0, 0, 0, 0, 0 ], [ 0, (z)^5,0,0, 0\
, 0 ],[ 0, 0, -(z)^2-(z)^5, 0, 0, 0 ], [ 0, 0, 0, (z)^4, 0, 0 ],[0,0, 0, 0, -(\
z)^4-(z)^7, 0 ], [ 0, 0, 0, 0, 0, (z)^7 ] ],[ [ z^3, 0, 0, 0,0, 0 ], [ 0, z^3,\
 0, 0, 0, 0 ],[ 0, 0, (z^3), 0, 0, 0 ], [ 0, 0, 0, (z^3)^2,0,0],[ 0, 0, 0, 0, \
(z^3)^2, 0 ], [ 0, 0, 0, 0, 0, (z^3)^2 ] ] ]
> ];
> G:=SmallGroup(54,6); 
> CCLR:=Classes(G);
> CCLR:=[CCLR[i][3]: i in [1..#CCLR]];
> T:=CharacterTable(G);
> D:=[LookupCharacter(T,CharacterRing(G)!char(CCLR,rho(G,K,L[i]))) : i in [1..\
#L]];
> D;
[ 1, 2, 4, 6, 5, 3, 7, 9, 8, 10 ]

This tells us that the sixth and tenth representations in GAP have characters \( \chi_3\) and \( \chi_{10}\) with respect to the Magma character table. Thus, the matrix generators for the representation \( \chi_3 + \chi_{10}\) are: 

> MG:=[Matrix([[-1,0,0,0,0,0,0],[ 0, 0, 0, 0, 0, 0, 1 ],[ 0, 0, 0, 0, 1, 0, 0]\
,[ 0, 0, 0, 0, 0, 1, 0 ],[ 0, 0, 1, 0, 0, 0, 0],[ 0, 0, 0, 1, 0, 0, 0 ], [0,1,\
 0, 0, 0, 0, 0 ]]), Matrix([[(z^3)^2,0,0,0,0,0,0],[ 0, 0, 0, 1, 0, 0,0 ],[0,1,\
 0, 0, 0, 0, 0 ],[ 0,0, 1, 0, 0, 0, 0 ],[ 0, 0, 0, 0, 0, 0, 1 ],[ 0, 0,0,0,1, \
0, 0],[ 0, 0, 0, 0, 0, 1, 0 ]]), Matrix([[1,0,0,0,0,0,0],[ 0, (z)^2, 0, 0, 0, \
0, 0 ], [ 0, 0, (z)^5,0,0,0,0], [ 0, 0, 0, -(z)^2-(z)^5, 0, 0, 0 ],[ 0, 0, 0, \
0, (z)^4, 0, 0 ],[0,0,0,0,0,-(z)^4-(z)^7, 0 ], [0, 0, 0, 0,0, 0, (z)^7 ]]), Ma\
trix([[1,0,0,0,0,0,0],[0, z^3, 0,0,0,0,0],[0, 0, z^3, 0, 0, 0, 0 ],[0,0,0,(z^3\
),0,0,0],[ 0, 0, 0, 0,(z^3)^2,0,0],[ 0, 0, 0, 0, 0, (z^3)^2, 0 ], [0,0,0,0,0,0\
,(z^3)^2 ]])];
> MG;
[
    [-1  0  0  0  0  0  0]
    [ 0  0  0  0  0  0  1]
    [ 0  0  0  0  1  0  0]
    [ 0  0  0  0  0  1  0]
    [ 0  0  1  0  0  0  0]
    [ 0  0  0  1  0  0  0]
    [ 0  1  0  0  0  0  0],

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

    [         1          0          0          0          0          0          0]
    [         0        z^2          0          0          0          0          0]
    [         0          0        z^5          0          0          0           0]
    [         0          0          0 -z^5 - z^2          0          0          0]
    [         0          0          0          0        z^4          0              0]
    [         0          0          0          0          0          z              0]
    [         0          0          0          0          0          0   -z^4 -         z],

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

Therefore, we obtain the following matrix surface kernel generators: 

> GL7K:=GeneralLinearGroup(7,K);
> MG:=[GL7K!MG[i] : i in [1..#MG]];
> rho:=hom< G -> GL7K | MG>;
> A:=rho(G.1);                     
> B:=rho(G.1 * G.2 * G.3^2 * G.4);
> C:=rho(G.2^2 * G.3 * G.4^2);
> Order(A);
2
> Order(B);
6
> Order(C);
9
> A;
[-1  0  0  0  0  0  0]
[ 0  0  0  0  0  0  1]
[ 0  0  0  0  1  0  0]
[ 0  0  0  0  0  1  0]
[ 0  0  1  0  0  0  0]
[ 0  0  0  1  0  0  0]
[ 0  1  0  0  0  0  0]
> B;
[   z^3 + 1          0          0          0          0          0          0]
[         0          0          0          0          0 -z^5 - z^2          0]
[         0          0          0          0          0          0        z^2]
[         0          0          0          0        z^5          0          0]
[         0   -z^4 - z          0          0          0          0          0]
[         0          0        z^4          0          0          0          0]
[         0          0          0          z          0          0          0]
> C;
[       z^3          0          0          0          0          0          0]
[         0          0        z^2          0          0          0          0]
[         0          0          0        z^5          0          0          0]
[         0 -z^5 - z^2          0          0          0          0          0]
[         0          0          0          0          0        z^4          0]
[         0          0          0          0          0          0          z]
[         0          0          0          0   -z^4 - z          0          0]

We use these matrix surface kernel generators to obtain candidate polynomials. 

> MatrixGens,SKG,Q:=RunGivenSKMatrixGenerators(54,7,[A,B,C]);
Set seed to 0.


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


--------------------------------------------------
Class |   1  2  3      4      5   6   7   8   9 10
Size  |   1  9  2      3      3   9   9   6   6  6
Order |   1  2  3      3      3   6   6   9   9  9
--------------------------------------------------
p  =  2   1  1  3      5      4   4   5   9   8 10
p  =  3   1  2  1      1      1   2   2   3   3  3
--------------------------------------------------
X.1   +   1  1  1      1      1   1   1   1   1  1
X.2   +   1 -1  1      1      1  -1  -1   1   1  1
X.3   0   1 -1  1      J   -1-J 1+J  -J   J-1-J  1
X.4   0   1  1  1      J   -1-J-1-J   J   J-1-J  1
X.5   0   1 -1  1   -1-J      J  -J 1+J-1-J   J  1
X.6   0   1  1  1   -1-J      J   J-1-J-1-J   J  1
X.7   +   2  0  2      2      2   0   0  -1  -1 -1
X.8   0   2  0  2 -2-2*J    2*J   0   0 1+J  -J -1
X.9   0   2  0  2    2*J -2-2*J   0   0  -J 1+J -1
X.10  +   6  0 -3      0      0   0   0   0   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 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]

[2]     Order 2       Length 9      
        Rep [-1  0  0  0  0  0  0]
        [ 0  0  0  0  0  0  1]
        [ 0  0  0  0  1  0  0]
        [ 0  0  0  0  0  1  0]
        [ 0  0  1  0  0  0  0]
        [ 0  0  0  1  0  0  0]
        [ 0  1  0  0  0  0  0]

[3]     Order 3       Length 2      
        Rep [      1       0       0       0       0       0       0]
        [      0    -z^9       0       0       0       0       0]
        [      0       0    -z^9       0       0       0       0]
        [      0       0       0    -z^9       0       0       0]
        [      0       0       0       0 z^9 - 1       0       0]
        [      0       0       0       0       0 z^9 - 1       0]
        [      0       0       0       0       0       0 z^9 - 1]

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

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

[6]     Order 6       Length 9      
        Rep [-z^9 + 1 0 0 0 0 0 0]
        [0 0 0 0 z^12 0 0]
        [0 0 0 0 0 -z^3 0]
        [0 0 0 0 0 0 -z^12 + z^3]
        [0 0 0 z^15 - z^6 0 0 0]
        [0 z^6 0 0 0 0 0]
        [0 0 -z^15 0 0 0 0]

[7]     Order 6       Length 9      
        Rep [z^9 0 0 0 0 0 0]
        [0 0 0 0 0 -z^12 + z^3 0]
        [0 0 0 0 0 0 z^12]
        [0 0 0 0 -z^3 0 0]
        [0 -z^15 0 0 0 0 0]
        [0 0 z^15 - z^6 0 0 0 0]
        [0 0 0 z^6 0 0 0]

[8]     Order 9       Length 6      
        Rep [-z^9 0 0 0 0 0 0]
        [0 0 0 -z^15 0 0 0]
        [0 z^15 - z^6 0 0 0 0 0]
        [0 0 z^6 0 0 0 0]
        [0 0 0 0 0 0 -z^3]
        [0 0 0 0 -z^12 + z^3 0 0]
        [0 0 0 0 0 z^12 0]

[9]     Order 9       Length 6      
        Rep [z^9 - 1 0 0 0 0 0 0]
        [0 0 z^12 0 0 0 0]
        [0 0 0 -z^3 0 0 0]
        [0 -z^12 + z^3 0 0 0 0 0]
        [0 0 0 0 0 z^15 - z^6 0]
        [0 0 0 0 0 0 z^6]
        [0 0 0 0 -z^15 0 0]

[10]    Order 9       Length 6      
        Rep [1 0 0 0 0 0 0]
        [0 z^6 0 0 0 0 0]
        [0 0 -z^15 0 0 0 0]
        [0 0 0 z^15 - z^6 0 0 0]
        [0 0 0 0 z^12 0 0]
        [0 0 0 0 0 -z^3 0]
        [0 0 0 0 0 0 -z^12 + z^3]


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 ]
S_1      =[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1 ]
I_2      =[ 1, 0, 0, 1, 0, 0, 0, 1, 0, 1 ]
S_2      =[ 1, 0, 0, 2, 0, 1, 1, 1, 1, 3 ]
H^0(C,2K)=[ 0, 0, 0, 1, 0, 1, 1, 0, 1, 2 ]
I_3      =[ 1, 2, 1, 1, 1, 0, 2, 2, 2, 6 ]
S_3      =[ 1, 3, 2, 1, 2, 1, 4, 3, 3, 9 ]
H^0(C,3K)=[ 0, 1, 1, 0, 1, 1, 2, 1, 1, 3 ]
I2timesS1=[ 1, 2, 1, 1, 2, 1, 2, 2, 3, 8 ]
Is clearly not generated by quadrics? false
Matrix Surface Kernel Generators:
Field K Cyclotomic Field of order 54 and degree 18
[
    [-1  0  0  0  0  0  0]
    [ 0  0  0  0  0  0  1]
    [ 0  0  0  0  1  0  0]
    [ 0  0  0  0  0  1  0]
    [ 0  0  1  0  0  0  0]
    [ 0  0  0  1  0  0  0]
    [ 0  1  0  0  0  0  0],

    [z^9 0 0 0 0 0 0]
    [0 0 0 0 0 -z^12 + z^3 0]
    [0 0 0 0 0 0 z^12]
    [0 0 0 0 -z^3 0 0]
    [0 -z^15 0 0 0 0 0]
    [0 0 z^15 - z^6 0 0 0 0]
    [0 0 0 z^6 0 0 0],

    [z^9 - 1 0 0 0 0 0 0]
    [0 0 z^12 0 0 0 0]
    [0 0 0 -z^3 0 0 0]
    [0 -z^12 + z^3 0 0 0 0 0]
    [0 0 0 0 0 z^15 - z^6 0]
    [0 0 0 0 0 0 z^6]
    [0 0 0 0 -z^15 0 0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 1
Multiplicity 1
[
    x_1*x_6 + x_2*x_4 + x_3*x_5
]
I2 contains a 1-dimensional subspace of CharacterRow 4
Dimension 2
Multiplicity 2
[
    x_0^2,
    x_1*x_6 - z^9*x_2*x_4 + (z^9 - 1)*x_3*x_5
]
I2 contains a 2-dimensional subspace of CharacterRow 8
Dimension 2
Multiplicity 1
[
    x_1*x_4 + (z^9 - 1)*x_2*x_5 - z^9*x_3*x_6,
    x_1*x_5 + (z^9 - 1)*x_2*x_6 - z^9*x_3*x_4
]
I2 contains a 6-dimensional subspace of CharacterRow 10
Dimension 18
Multiplicity 3
[
    x_0*x_1,
    x_0*x_2,
    x_0*x_3,
    x_0*x_4,
    x_0*x_5,
    x_0*x_6,
    x_1^2,
    x_1*x_2,
    x_1*x_3,
    x_2^2,
    x_2*x_3,
    x_3^2,
    x_4^2,
    x_4*x_5,
    x_4*x_6,
    x_5^2,
    x_5*x_6,
    x_6^2
]

The first isotypical subspace, which corresponds to the character \( \chi_1\) in the character table shown above, yields the polynomial \[ x_1 x_6 + x_2 x_4 + x_3 x_5 \]

The second isotypical subspace, which corresponds to the character \( \chi_4\) in the character table shown above, yields a polynomial of the form \[ c_1(x_0^2) + c_2(x_1 x_6 - \zeta_{6} x_2 x_4 + (\zeta_{6} - 1) x_3 x_5) \] Assume that \( c_1 \) and \(c_2\) are nonzero. Then by scaling \( x_0\), we may assume that \( c_1 = c_2 =1\).

The third isotypical subspace, which corresponds to the character \( \chi_8\) in the character table shown above, yields the polynomials \[ \begin{array}{l} x_1 x_4 + (\zeta_{6} - 1) x_2 x_5 - \zeta_{6} x_3 x_6\\ x_1 x_5 + (\zeta_{6} - 1) x_2 x_6 - \zeta_{6} x_3 x_4 \end{array} \]

The fourth isotypical subspace corresponds to the character \( \chi_{10}\) in the character table shown above. Note that the matrix surface kernel generators have a block diagonal form with blocks of size \(1 \times 1\) and \(6 \times 6\). Also, the matrix surface kernel generators only permute and scale the variables \(x_0,\ldots,x_6\); there is exactly one nonzero entry in each row and each column. We therefore let \( \operatorname{Span} \langle x_0x_1, x_0x_2, x_0x_3, x_0x_4, x_0x_5, x_0x_6 \rangle \) generate one copy of \(V_{10}\). We use the FindParallelBases function to find ordered bases of \( \operatorname{Span}\langle x_1^2, x_2^2, x_3^2, x_4^2, x_5^2, x_6^2 \rangle \) and \( \operatorname{Span}\langle x_1x_2, x_1x_3, x_2x_3, x_4x_5, x_4x_6, x_5x_6 \rangle \), so that the action of \( G\) is given by the same matrices relative to all three ordered bases. This produces: 

> GL7K:=Parent(MatrixGens[1]);
> MatrixG:=sub<GL7K | MatrixGens>;
> FindParallelBases(MatrixG,[Q[4][1],Q[4][2],Q[4][3],Q[4][4],Q[4][5],Q[4][6]],\
[Q[4][7],Q[4][10],Q[4][12],Q[4][13],Q[4][16],Q[4][18]]);
[       z^9*x_5^2]
[          -x_6^2]
[(-z^9 + 1)*x_4^2]
[           x_1^2]
[ (z^9 - 1)*x_2^2]
[      -z^9*x_3^2]
> FindParallelBases(MatrixG,[Q[4][1],Q[4][2],Q[4][3],Q[4][4],Q[4][5],Q[4][6]],\
[Q[4][8],Q[4][9],Q[4][11],Q[4][14],Q[4][15],Q[4][17]]);
[          -x_4*x_6]
[(-z^9 + 1)*x_4*x_5]
[       z^9*x_5*x_6]
[ (z^9 - 1)*x_2*x_3]
[      -z^9*x_1*x_3]
[           x_1*x_2]
Therefore, the candidate polynomials for the subspace are: \[ \begin{array}{l} c_3(x_0x_1)+c_4(\zeta_{6}x_5^2)+c_5(-x_4x_6), \\ c_3(x_0x_2)+c_4(-x_6^2)+c_5((-\zeta_{6} + 1)x_4x_5), \\ c_3(x_0x_3)+c_4((-\zeta_{6}+ 1)x_4^2)+c_5(\zeta_{6}x_5x_6), \\ c_3(x_0x_4)+c_4(x_1^2)+c_5((\zeta_{6} - 1)x_2x_3), \\ c_3(x_0x_5)+c_4((\zeta_{6}- 1)x_2^2)+c_5(-\zeta_{6}x_1x_3), \\ c_3(x_0x_6)+c_4(-\zeta_{6}x_3^2)+c_5(x_1x_2); \end{array} \] Assume that \(c_3, c_4, c_5\) are nonzero. Then after dividing by \(c_3\) we may assume that \( c_3 = 1\).

For generic values of \(c_4\), \(c_5\), the intersection of these 10 quadrics is empty: 

> K<z_6>:=CyclotomicField(6);
> c_4:=1;
> c_5:=1;
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[x_1*x_6 + x_2*x_4 + x_3*x_5,
> x_0^2+x_1*x_6 - z_6*x_2*x_4 +(z_6 - 1)*x_3*x_5,
> x_1*x_4 + (z_6 - 1)*x_2*x_5 - z_6*x_3*x_6,
> x_1*x_5+(z_6 -1)*x_2*x_6 - z_6*x_3*x_4,
> x_0*x_1+c_4*(z_6*x_5^2)+c_5*(-x_4*x_6),
> x_0*x_2+c_4*(-x_6^2)+c_5*((-z_6 + 1)*x_4*x_5),
> x_0*x_3+c_4*((-z_6 + 1)*x_4^2)+c_5*(z_6*x_5*x_6),
> x_0*x_4+c_4*(x_1^2)+c_5*((z_6 -1)*x_2*x_3),
> x_0*x_5+c_4*((z_6 - 1)*x_2^2)+c_5*(-z_6*x_1*x_3),
> x_0*x_6+c_4*(-z_6*x_3^2)+c_5*(x_1*x_2)]);
> Dimension(X);
-1
We turn to Macaulay2 to compute part of a flattening stratification to find special coefficient values that yield a smooth curve.

Flattening stratification in Macaulay2

Macaulay2, version 1.7

i1 : loadPackage("Cyclotomic");

i2 : K=cyclotomicField(6);

i3 : z_6=K_0;

i4 : S=K[c_4,c_5,Degrees=>{0,0}];

i5 : T=S[x_0..x_6];

i6 : I=ideal {x_1*x_6 + x_2*x_4 + x_3*x_5,
     x_0^2+x_1*x_6 - z_6*x_2*x_4 +(z_6 - 1)*x_3*x_5,
     x_1*x_4 + (z_6 - 1)*x_2*x_5 - z_6*x_3*x_6,
     x_1*x_5+(z_6 -1)*x_2*x_6 - z_6*x_3*x_4,
     x_0*x_1+c_4*(z_6*x_5^2)+c_5*(-x_4*x_6),
     x_0*x_2+c_4*(-x_6^2)+c_5*((-z_6 + 1)*x_4*x_5),
     x_0*x_3+c_4*((-z_6 + 1)*x_4^2)+c_5*(z_6*x_5*x_6),
     x_0*x_4+c_4*(x_1^2)+c_5*((z_6 -1)*x_2*x_3),
     x_0*x_5+c_4*((z_6 - 1)*x_2^2)+c_5*(-z_6*x_1*x_3),
     x_0*x_6+c_4*(-z_6*x_3^2)+c_5*(x_1*x_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} => 13
           {4, 0} => 1

o8 : Tally

i9 : for i from 0 to #L-1 do (if degree(L_i) == {3,0} then print toString(L_i) << endl)
(c_4+c_5)*x_1*x_5*x_6+(ww_6*c_4+ww_6*c_5)*x_2*x_6^2
(c_4+c_5)*x_4*x_5^2+((ww_6-1)*c_4+(ww_6-1)*c_5)*x_4^2*x_6+(-ww_6*c_4-ww_6*c_5)*x_5*x_6^2
(c_4+c_5)*x_2*x_5^2+((-ww_6+1)*c_4+(-ww_6+1)*c_5)*x_1*x_6^2
(c_4+c_5)*x_1*x_5^2+(ww_6*c_4+ww_6*c_5)*x_2*x_5*x_6
(c_4+c_5)*x_4^2*x_5+((ww_6-1)*c_4+(ww_6-1)*c_5)*x_5^2*x_6+(-ww_6*c_4-ww_6*c_5)*x_4*x_6^2
(c_4+c_5)*x_3^2*x_5+ww_6*c_5*x_2^2*x_6+c_5*x_1*x_3*x_6+(ww_6-1)*x_0*x_5*x_6
x_2*x_3*x_5+((1/3)*ww_6+1/3)*c_5*x_0*x_4*x_5+(-ww_6+1)*x_1*x_2*x_6+(-(2/3)*ww_6+1/3)*c_4*x_0*x_6^2
x_1*x_3*x_5+((1/3)*ww_6-2/3)*c_4*x_0*x_5^2+(-ww_6+1)*x_1^2*x_6+(-(2/3)*ww_6+1/3)*c_5*x_0*x_4*x_6
x_2^2*x_5+((1/3)*ww_6+1/3)*c_4*x_0*x_5^2+(ww_6-1)*x_1^2*x_6+(ww_6-1)*x_2*x_3*x_6+((1/3)*ww_6-2/3)*c_5*x_0*x_4*x_6
x_1*x_2*x_5+ww_6*x_3^2*x_5+(ww_6-1)*x_2^2*x_6+ww_6*x_1*x_3*x_6
x_1^2*x_5+((1/3)*ww_6+1/3)*c_5*x_0*x_4*x_5+(-ww_6+1)*x_3^2*x_6+(-(2/3)*ww_6+1/3)*c_4*x_0*x_6^2
c_4*x_4^3-ww_6*c_4*x_5^3+(ww_6-1)*c_4*x_6^3
x_0*x_4^2+(-ww_6+2)*c_5*x_3^2*x_5+ww_6*c_5*x_2^2*x_6+((ww_6-1)*c_4+(-ww_6+1)*c_5)*x_1*x_3*x_6
The leading coefficients of the generators above suggest the choice \( c_{5}= -c_{4}\). We rerun the calculation with this choice. 
i10 : S=K[c_4,Degrees=>{0}];

i11 : c_5=-c_4;

i12 : T=S[x_0..x_6];

i13 : I=ideal {x_1*x_6 + x_2*x_4 + x_3*x_5,
      x_0^2+x_1*x_6 - z_6*x_2*x_4 +(z_6 - 1)*x_3*x_5,
      x_1*x_4 + (z_6 - 1)*x_2*x_5 - z_6*x_3*x_6,
      x_1*x_5+(z_6 -1)*x_2*x_6 - z_6*x_3*x_4,
      x_0*x_1+c_4*(z_6*x_5^2)+c_5*(-x_4*x_6),
      x_0*x_2+c_4*(-x_6^2)+c_5*((-z_6 + 1)*x_4*x_5),
      x_0*x_3+c_4*((-z_6 + 1)*x_4^2)+c_5*(z_6*x_5*x_6),
      x_0*x_4+c_4*(x_1^2)+c_5*((z_6 -1)*x_2*x_3),
      x_0*x_5+c_4*((z_6 - 1)*x_2^2)+c_5*(-z_6*x_1*x_3),
      x_0*x_6+c_4*(-z_6*x_3^2)+c_5*(x_1*x_2)
      };

o13 : Ideal of T
										  
i14 : L=flatten entries gens gb I
										  
i15 : for i from 0 to #L-1 do print toString(L_i) << endl
x_3*x_4+(ww_6-1)*x_1*x_5-ww_6*x_2*x_6
x_2*x_4+x_3*x_5+x_1*x_6
x_1*x_4+(ww_6-1)*x_2*x_5-ww_6*x_3*x_6
x_0*x_3+(-ww_6+1)*c_4*x_4^2-ww_6*c_4*x_5*x_6
c_4*x_2^2+(-ww_6+1)*c_4*x_1*x_3-ww_6*x_0*x_5
c_4*x_1*x_2+ww_6*c_4*x_3^2-x_0*x_6
x_0*x_2+(ww_6-1)*c_4*x_4*x_5-c_4*x_6^2
c_4*x_1^2+(-ww_6+1)*c_4*x_2*x_3+x_0*x_4
x_0*x_1+ww_6*c_4*x_5^2+c_4*x_4*x_6
x_0^2+(2*ww_6-1)*x_3*x_5+(ww_6+1)*x_1*x_6
(c_4^2+1)*x_3*x_5*x_6+((-ww_6+1)*c_4^2-ww_6+1)*x_1*x_6^2
(c_4^2+1)*x_3*x_5^2+((-ww_6+1)*c_4^2-ww_6+1)*x_1*x_5*x_6
(c_4^2+1)*x_1*x_5^2+(ww_6*c_4^2+ww_6)*x_3*x_6^2
(c_4^2+1)*x_0*x_5^2+((-ww_6+1)*c_4^2-ww_6+1)*x_0*x_4*x_6
(c_4^2+1)*x_0*x_4*x_5+(ww_6*c_4^2+ww_6)*x_0*x_6^2
(c_4^2+1)*x_3^2*x_5+((-ww_6+1)*c_4^2-ww_6+1)*x_1*x_3*x_6
(output abbreviated)
The leading coefficients of the generators above suggest the choice \( c_{4}= i\).

We noticed that scaling \(x_0\) by \(i\) would allow us to write these equations over a smaller field. This finally yields the equations 


x_1*x_6 + x_2*x_4 + x_3*x_5,
x_0^2-x_1*x_6 + z_6*x_2*x_4 - (z_6 - 1)*x_3*x_5,
x_1*x_4 + (z_6 - 1)*x_2*x_5 - z_6*x_3*x_6,
x_1*x_5+(z_6 -1)*x_2*x_6 - z_6*x_3*x_4,
x_0*x_1+z_6*x_5^2+x_4*x_6,
x_0*x_2-x_6^2+(z_6 -1)*x_4*x_5,
x_0*x_3-(z_6 - 1)*x_4^2-z_6*x_5*x_6,
x_0*x_4+x_1^2-(z_6 -1)*x_2*x_3,
x_0*x_5+(z_6 - 1)*x_2^2+z_6*x_1*x_3,
x_0*x_6-z_6*x_3^2-x_1*x_2
In the next section, we show that this yields a smooth genus 7 curve with the desired automorphisms.

Checking the equations in Magma

> K<z_18>:=CyclotomicField(18);
> z_9:=z_18^2;
> z_6:=z_18^3;
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[x_1*x_6 + x_2*x_4 + x_3*x_5,
> x_0^2-x_1*x_6 + z_6*x_2*x_4 - (z_6 - 1)*x_3*x_5,
> x_1*x_4 + (z_6 - 1)*x_2*x_5 - z_6*x_3*x_6,
> x_1*x_5+(z_6 -1)*x_2*x_6 - z_6*x_3*x_4,
> x_0*x_1+z_6*x_5^2+x_4*x_6,
> x_0*x_2-x_6^2+(z_6 -1)*x_4*x_5,
> x_0*x_3-(z_6 - 1)*x_4^2-z_6*x_5*x_6,
> x_0*x_4+x_1^2-(z_6 -1)*x_2*x_3,
> x_0*x_5+(z_6 - 1)*x_2^2+z_6*x_1*x_3,
> x_0*x_6-z_6*x_3^2-x_1*x_2]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> GL7K:=GeneralLinearGroup(7,K);
> A:=Matrix([
> [-1,0,0,0,0,0,0],
> [0,0,0,0,0,0,1],
> [0,0,0,0,1,0,0],
> [0,0,0,0,0,1,0],
> [0,0,1,0,0,0,0],
> [0,0,0,1,0,0,0],
> [0,1,0,0,0,0,0]
> ]);
> B:=Matrix([
> [z_9^3+1,0,0,0,0,0,0],
> [0,0,0,0,0,-z_9^5 - z_9^2 ,0],
> [0,0,0,0,0,0,z_9^2],
> [0,0,0,0,z_9^5,0,0],
> [0,-z_9^4 - z_9,0,0,0,0,0],
> [0,0,z_9^4,0,0,0,0],
> [0,0,0,z_9,0,0,0]
> ]);
> IdentifyGroup(sub<GL7K | A,B>);
<54, 6>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_6
x_4
x_5
x_2
x_3
x_1
and inverse
-x_0
x_6
x_4
x_5
x_2
x_3
x_1
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
z_18^3*x_0
-z_18^5*x_4
(z_18^5 - z_18^2)*x_5
z_18^2*x_6
-z_18*x_3
(-z_18^4 + z_18)*x_1
z_18^4*x_2
and inverse
(-z_18^3 + 1)*x_0
z_18^2*x_5
-z_18^5*x_6
(z_18^5 - z_18^2)*x_4
z_18^4*x_1
-z_18*x_2
(-z_18^4 + z_18)*x_3

Computing the Betti table in Macaulay2

We use Macaulay2 to compute the Betti table of the ideal generated by these equations. 
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : loadPackage("Cyclotomic");

i2 : K=cyclotomicField(6);

i3 : z_6=K_0;

i4 : R=K[x_0..x_6];

i5 : I=ideal {x_1*x_6 + x_2*x_4 + x_3*x_5,
     x_0^2-x_1*x_6 + z_6*x_2*x_4 - (z_6 - 1)*x_3*x_5,
     x_1*x_4 + (z_6 - 1)*x_2*x_5 - z_6*x_3*x_6,
     x_1*x_5+(z_6 -1)*x_2*x_6 - z_6*x_3*x_4,
     x_0*x_1+z_6*x_5^2+x_4*x_6,
     x_0*x_2-x_6^2+(z_6 -1)*x_4*x_5,
     x_0*x_3-(z_6 - 1)*x_4^2-z_6*x_5*x_6,
     x_0*x_4+x_1^2-(z_6 -1)*x_2*x_3,
     x_0*x_5+(z_6 - 1)*x_2^2+z_6*x_1*x_3,
     x_0*x_6-z_6*x_3^2-x_1*x_2};

o5 : Ideal of R

i6 : betti res I

            0  1  2  3  4 5
o6 = total: 1 10 16 16 10 1
         0: 1  .  .  .  . .
         1: . 10 16  .  . .
         2: .  .  . 16 10 .
         3: .  .  .  .  . 1

o6 : BettiTally

o6 : BettiTally
By [Schreyer1986], this Betti table implies that the curve is general, in the sense of that paper.

The second Riemann surface

We begin to analyze Breuer's second set of surface kernel generators. 

> RunGivenSKG(G,7,[G.1, G.1*G.2^2*G.3^2*G.4^2, G.2*G.3 ]);
Set seed to 0.


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


--------------------------------------------------
Class |   1  2  3      4      5   6   7   8  9  10
Size  |   1  9  2      3      3   9   9   6  6   6
Order |   1  2  3      3      3   6   6   9  9   9
--------------------------------------------------
p  =  2   1  1  3      5      4   4   5  10  9   8
p  =  3   1  2  1      1      1   2   2   3  3   3
--------------------------------------------------
X.1   +   1  1  1      1      1   1   1   1  1   1
X.2   +   1 -1  1      1      1  -1  -1   1  1   1
X.3   0   1 -1  1      J   -1-J 1+J  -J-1-J  1   J
X.4   0   1  1  1   -1-J      J   J-1-J   J  1-1-J
X.5   0   1  1  1      J   -1-J-1-J   J-1-J  1   J
X.6   0   1 -1  1   -1-J      J  -J 1+J   J  1-1-J
X.7   +   2  0  2      2      2   0   0  -1 -1  -1
X.8   0   2  0  2    2*J -2-2*J   0   0 1+J -1  -J
X.9   0   2  0  2 -2-2*J    2*J   0   0  -J -1 1+J
X.10  +   6  0 -3      0      0   0   0   0  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 9      
        Rep G.1

[3]     Order 3       Length 2      
        Rep G.4

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

[5]     Order 3       Length 3      
        Rep G.2

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

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

[8]     Order 9       Length 6      
        Rep G.2 * G.3 * G.4

[9]     Order 9       Length 6      
        Rep G.3

[10]    Order 9       Length 6      
        Rep G.2^2 * G.3 * G.4^2


Surface kernel generators:  [ G.1, G.1 * G.2^2 * G.3^2 * G.4^2, G.2 * G.3 ]
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 ]
S_1      =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ]
I_2      =[ 1, 0, 0, 0, 1, 0, 0, 0, 1, 1 ]
S_2      =[ 1, 0, 0, 1, 2, 0, 1, 1, 1, 3 ]
H^0(C,2K)=[ 0, 0, 0, 1, 1, 0, 1, 1, 0, 2 ]
I_3      =[ 1, 2, 1, 0, 1, 1, 2, 2, 2, 6 ]
S_3      =[ 1, 3, 2, 1, 1, 2, 4, 3, 3, 9 ]
H^0(C,3K)=[ 0, 1, 1, 1, 0, 1, 2, 1, 1, 3 ]
I2timesS1=[ 1, 2, 1, 1, 1, 2, 2, 3, 2, 8 ]
Is clearly not generated by quadrics? false
No subgroup found

RunGivenSKG(
    G: GrpPC : G,
    genus: 7,
    SKG: [ G.1, G.1 * G.2^2 * G.3^2 * G.4^2, G.2 * G.3 ]
)
FindMatrixGenerators(
    G: GrpPC : G,
    genus: 7,
    T:   Character Table of Group G --------------------------   --...,
    CCL: Conjugacy Classes of group G ---------------------------- [1...,
    M: [ G.1, G.1 * G.2^2 * G.3^2 * G.4^2, G.2 * G.3 ]
)
In file "autcv10c.txt", line 220, column 28:
>>       ags:=ActionGenerators(GModule(T[i]));
                              ^
Runtime error in 'ActionGenerators': Bad argument types
Argument types given: BoolElt
The error "No subgroup found" tells us that Magma has an internal error when finding the matrix generators of representations with character \( \chi_{6}\) and \( \chi_{10}\).

However, from the character table shows that \( \chi_{6} = \overline{ \chi_3} \). Thus, we may use the complex conjugates of the matrix surface kernel generators found above for the first Riemann surface as surface kernel generators for the second Riemann surface, and the equations of the second Riemann surface may be taken as the complex conjugates of the equations of the first.