Fordham University

Finding equations of a 1-parameter family of genus 7 Riemann surfaces with 32 automorphisms

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 (32,43) in the GAP library of small groups. The quotient of any member of this family by its automorphism group has genus zero, and the quotient morphism is branched over four points with ramification indices (2,2,2,8).

We use GAP, Macaulay2 and 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 autcv10.txt used below. 
Magma V2.20-5     Sat Nov 14 2015 15:57:20 on ace-math01 [Seed = 3603374323]
Type ? for help.  Type -D to quit.
> load "/app/home/dswinarski/autcv10.txt";
Loading "/app/home/dswinarski/autcv10.txt"
> G:=SmallGroup(32,43);
> MatrixG,MatrixSKG,Q,C:=RunExample(G,7,[2,2,2,8]);
Set seed to 0.


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


-----------------------------------------
Class |   1  2  3  4  5  6  7  8  9 10 11
Size  |   1  1  2  4  4  4  2  2  4  4  4
Order |   1  2  2  2  2  2  4  4  4  8  8
-----------------------------------------
p  =  2   1  1  1  1  1  1  2  2  2  7  7
-----------------------------------------
X.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
X.3   +   1  1 -1  1 -1 -1  1 -1  1  1 -1
X.4   +   1  1 -1 -1 -1  1  1 -1  1 -1  1
X.5   +   1  1 -1  1  1 -1  1 -1 -1 -1  1
X.6   +   1  1 -1 -1  1  1  1 -1 -1  1 -1
X.7   +   1  1  1  1 -1  1  1  1 -1 -1 -1
X.8   +   1  1  1 -1 -1 -1  1  1 -1  1  1
X.9   +   2  2  2  0  0  0 -2 -2  0  0  0
X.10  +   2  2 -2  0  0  0 -2  2  0  0  0
X.11  +   4 -4  0  0  0  0  0  0  0  0  0



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

[4]     Order 2       Length 4      
        Rep G.2

[5]     Order 2       Length 4      
        Rep G.1

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

[7]     Order 4       Length 2      
        Rep G.4

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

[9]     Order 4       Length 4      
        Rep G.1 * G.3

[10]    Order 8       Length 4      
        Rep G.1 * G.2 * G.3

[11]    Order 8       Length 4      
        Rep G.1 * G.2


Surface kernel generators:  [ G.3 * G.5, G.1 * G.4, G.2 * G.4, 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 ]
S_1      =[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1 ]
H^0(C,K) =[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1 ]
I_2      =[ 2, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1 ]
S_2      =[ 3, 1, 0, 0, 1, 1, 2, 0, 3, 1, 3 ]
H^0(C,2K)=[ 1, 1, 0, 0, 1, 0, 1, 0, 2, 1, 2 ]
I_3      =[ 1, 0, 2, 3, 1, 3, 1, 1, 2, 5, 7 ]
S_3      =[ 1, 1, 3, 5, 2, 4, 2, 2, 3, 7, 11 ]
H^0(C,3K)=[ 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 4 ]
I2timesS1=[ 1, 1, 2, 4, 2, 3, 2, 1, 3, 6, 9 ]
Is clearly not generated by quadrics? false
No subgroup found

RunExample(
    G: GrpPC : G,
    genus: 7,
    E: [ 2, 2, 2, 8 ]
)
FindMatrixGenerators(
    G: GrpPC : G,
    genus: 7,
    T:   Character Table of Group G --------------------------   --...,
    CCL: Conjugacy Classes of group G ---------------------------- [1...,
    M: [ G.3 * G.5, G.1 * G.4, G.2 * G.4, G.1 * G.2 * G.3 * G.4 * G...
)
In file "/app/home/dswinarski/autcv10.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 \( \chi_{4},\) \( \chi_{10},\) and/or \( \chi_{11}\).

We obtain these matrices using GAP. First, we check that GAP and Magma use the same polycyclic generators for the group (32,43), by showing that the GAP relations are satisfied by the Magma group, and the Magma relations are satisfied by the GAP group.

Next, in GAP, we obtain matrix generators for the irreducible representations of this group: 


gap> IrreducibleRepresentations(SmallGroup(32,43));
[ Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], 
      [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], 
      [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ -1 ] ], 
      [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ -1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ -1 ] ], 
      [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ -1 ] ], [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ],
      [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], 
      [ [ -1, 0 ], [ 0, -1 ] ], [ [ -1, 0 ], [ 0, -1 ] ], 
      [ [ 1, 0 ], [ 0, 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ 0, 0, 0, 1 ], [ 0, 0, E(4), 0 ], [ 0, -E(4), 0, 0 ], [ 1, 0, 0, 0 ] 
         ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ]
        , [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ]
        , 
      [ [ E(4), 0, 0, 0 ], [ 0, -E(4), 0, 0 ], [ 0, 0, E(4), 0 ], 
          [ 0, 0, 0, -E(4) ] ], 
      [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ] 
     ] ]

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<z>:=CyclotomicField(32);
> i:=z^8;
> GL7K:=GeneralLinearGroup(7,K);
> 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 ] ], [ [ 1 ] ] ], 
> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ 1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ -1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ 1 ] ], [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ -1 ] ], [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ],[ \
[ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], 
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], [ [ -1, 0 ], [ 0, -1 ] ],\
 [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], 
> [ [ [ 0, 0, 0, 1 ], [ 0, 0, i, 0 ], [ 0, -i, 0, 0 ], [ 1, 0, 0, 0 ] ], [ [ 0\
, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ], [ [ 1, 0, 0, 0 \
], [ 0, 1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ], [ [ i, 0, 0, 0 ], [ 0, \
-i, 0, 0 ], [ 0, 0, i, 0 ], [ 0, 0, 0, -i ] ], [ [ -1, 0, 0, 0 ], [ 0, -1, 0, \
0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ] ]
> ];
> 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, 7, 2, 8, 5, 3, 6, 4, 9, 10, 11 ]

This tells us that the eighth, tenth, and eleventh representations in GAP have characters \( \chi_4\), \(\chi_{10}\), and \( \chi_{11}\) with respect to the Magma character table. Thus, the matrix generators for the representation \( \chi_4 + \chi_{10}+\chi_{11}\) are: 


> MG:=[
> 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, 0, 0, 1],
> [0, 0, 0, 0, 0, i, 0],
> [0, 0, 0, 0, -i, 0, 0],
> [0, 0, 0, 1, 0, 0, 0]
> ]),
> Matrix([
> [-1, 0, 0, 0, 0, 0, 0],
> [0, 1, 0, 0, 0, 0, 0],
> [0, 0, -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, 1],
> [0, 0, 0, 0, 0, 1, 0]
> ]),
> Matrix([
> [-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([
> [1, 0, 0, 0, 0, 0, 0],
> [0, -1, 0, 0, 0, 0, 0],
> [0, 0, -1, 0, 0, 0, 0],
> [0, 0, 0, i, 0, 0, 0],
> [0, 0, 0, 0, -i, 0, 0],
> [0, 0, 0, 0, 0, i, 0],
> [0, 0, 0, 0, 0, 0, -i]
> ]),
> Matrix([
> [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]
> ])
> ];
> MG:=[GL7K!MG[i]: i in [1..5]];
> rho:=hom< G -> GL7K | MG>;
> A:=rho(G.3 * G.5);
> B:=rho(G.1 * G.4);
> C:=rho(G.2 * G.4);
> D:=rho(G.1 * G.2 * G.3 * G.4 * G.5);
> A;
[-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]
> B;
[  -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    0    0 -z^8]
[   0    0    0    0    0   -1    0]
[   0    0    0    0   -1    0    0]
[   0    0    0  z^8    0    0    0]
> C;
[  -1    0    0    0    0    0    0]
[   0   -1    0    0    0    0    0]
[   0    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 -z^8]
[   0    0    0    0    0  z^8    0]
> D;
[ -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   0 z^8   0]
[  0   0   0   0   0   0   1]
[  0   0   0  -1   0   0   0]
[  0   0   0   0 z^8   0   0]

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


> MatrixG,MatrixSKG,Q,C:=RunGivenSKMatrixGenerators(32,7,[A,B,C,D]);
Set seed to 0.


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


-----------------------------------------
Class |   1  2  3  4  5  6  7  8  9 10 11
Size  |   1  1  2  4  4  4  2  2  4  4  4
Order |   1  2  2  2  2  2  4  4  4  8  8
-----------------------------------------
p  =  2   1  1  1  1  1  1  2  2  2  7  7
-----------------------------------------
X.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
X.3   +   1  1  1 -1  1 -1  1  1  1 -1 -1
X.4   +   1  1 -1 -1  1  1  1 -1 -1 -1  1
X.5   +   1  1  1  1 -1  1  1  1 -1 -1 -1
X.6   +   1  1 -1  1 -1 -1  1 -1  1 -1  1
X.7   +   1  1  1 -1 -1 -1  1  1 -1  1  1
X.8   +   1  1 -1 -1 -1  1  1 -1  1  1 -1
X.9   +   2  2  2  0  0  0 -2 -2  0  0  0
X.10  +   2  2 -2  0  0  0 -2  2  0  0  0
X.11  +   4 -4  0  0  0  0  0  0  0  0  0



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 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]

[3]     Order 2       Length 2      
        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]

[4]     Order 2       Length 4      
        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    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]

[5]     Order 2       Length 4      
        Rep [  -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    0    0 -z^8]
        [   0    0    0    0    0   -1    0]
        [   0    0    0    0   -1    0    0]
        [   0    0    0  z^8    0    0    0]

[6]     Order 2       Length 4      
        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    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]

[7]     Order 4       Length 2      
        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  z^8    0    0    0]
        [   0    0    0    0 -z^8    0    0]
        [   0    0    0    0    0  z^8    0]
        [   0    0    0    0    0    0 -z^8]

[8]     Order 4       Length 2      
        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  z^8    0    0    0]
        [   0    0    0    0 -z^8    0    0]
        [   0    0    0    0    0 -z^8    0]
        [   0    0    0    0    0    0  z^8]

[9]     Order 4       Length 4      
        Rep [   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    0    0 -z^8]
        [   0    0    0    0    0   -1    0]
        [   0    0    0    0    1    0    0]
        [   0    0    0 -z^8    0    0    0]

[10]    Order 8       Length 4      
        Rep [  -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    0 -z^8    0]
        [   0    0    0    0    0    0   -1]
        [   0    0    0    1    0    0    0]
        [   0    0    0    0 -z^8    0    0]

[11]    Order 8       Length 4      
        Rep [   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    0 -z^8    0]
        [   0    0    0    0    0    0   -1]
        [   0    0    0   -1    0    0    0]
        [   0    0    0    0  z^8    0    0]


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 ]
S_1      =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1 ]
I_2      =[ 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1 ]
S_2      =[ 3, 1, 1, 1, 2, 0, 0, 0, 3, 1, 3 ]
H^0(C,2K)=[ 1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 2 ]
I_3      =[ 1, 3, 0, 1, 1, 3, 1, 2, 2, 5, 7 ]
S_3      =[ 1, 4, 1, 2, 2, 5, 2, 3, 3, 7, 11 ]
H^0(C,3K)=[ 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4 ]
I2timesS1=[ 1, 3, 1, 2, 2, 4, 1, 2, 3, 6, 9 ]
Is clearly not generated by quadrics? false
Matrix Surface Kernel Generators:
Field K Cyclotomic Field of order 32 and degree 16
[
    [-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],

    [  -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    0    0 -z^8]
    [   0    0    0    0    0   -1    0]
    [   0    0    0    0   -1    0    0]
    [   0    0    0  z^8    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    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],

    [ -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   0 z^8   0]
    [  0   0   0   0   0   0   1]
    [  0   0   0  -1   0   0   0]
    [  0   0   0   0 z^8   0   0]
]
Finding quadrics:
I2 contains a 2-dimensional subspace of CharacterRow 1
Dimension 3
Multiplicity 3
[
    x_0^2,
    x_1^2 + x_2^2,
    x_3*x_4 - z^8*x_5*x_6
]
I2 contains a 1-dimensional subspace of CharacterRow 2
Dimension 1
Multiplicity 1
[
    x_3*x_6 - x_4*x_5
]
I2 contains a 1-dimensional subspace of CharacterRow 5
Dimension 2
Multiplicity 2
[
    x_1^2 - x_2^2,
    x_3*x_4 + z^8*x_5*x_6
]
I2 contains a 2-dimensional subspace of CharacterRow 9
Dimension 6
Multiplicity 3
[
    x_0*x_1,
    x_0*x_2,
    x_3^2,
    x_4^2,
    x_5^2,
    x_6^2
]
I2 contains a 4-dimensional subspace of CharacterRow 11
Dimension 12
Multiplicity 3
[
    x_0*x_3,
    x_0*x_4,
    x_0*x_5,
    x_0*x_6,
    x_1*x_3,
    x_1*x_4,
    x_1*x_5,
    x_1*x_6,
    x_2*x_3,
    x_2*x_4,
    x_2*x_5,
    x_2*x_6
]

The first isotypical subspace, which corresponds to the character \( \chi_1\) in the character table shown above, yields polynomials of the form 


c_1*(x_0^2)+c_2*(x_1^2 + x_2^2)+c_3*(x_3*x_4 - z^8*x_5*x_6)
c_4*(x_0^2)+c_5*(x_1^2 + x_2^2)+c_6*(x_3*x_4 - z^8*x_5*x_6)

Assume that \((c_1,c_2)\) and \((c_4,c_5)\) are linearly independent. After row reducing, we may assume that \((c_1,c_2)=(1,0)\) and \((c_4,c_5)=(0,1).\)

The second isotypical subspace, which corresponds to the character \( \chi_2\) in the character table shown above, yields the polynomial 


x_3*x_6 - x_4*x_5.

The third isotypical subspace, which corresponds to the character \( \chi_5\) in the character table shown above, yields a polynomial of the form 


c_7*(x_1^2 - x_2^2)+c_8*(x_3*x_4 + z^8*x_5*x_6)

Assume that \(c_7\) is nonzero. After dividing by \(c_7\), we may assume that \(c_7= 1.\)

The fourth 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 \(1 \times 1\), \(2 \times 2\), and \(4 \times 4\). We therefore let the first two polynomials shown here generate one copy of \(V_{9}\) and find two ordered bases of the span of the last four polynomials such that the action of \(G\) is given by the same matrices relative to all three ordered bases. This yields: 


c_9*(x_0*x_1)+c_10*(x_3^2-x_4^2)+c_11*(x_5^2-x_6^2),
c_9*(x_0*x_2)+c_10*(-x_5^2-x_6^2)+c_11*(x_3^2+x_4^2),

Assume that \(c_{9}\) and \(c_{10}\) are nonzero. After scaling \(x_0\) and dividing, we may assume that \(c_{9} = c_{10}= 1.\)

The fifth isotypical subspace corresponds to the character \( \chi_{11}\) in the character table shown above. Again we three ordered bases such that the action of \(G\) is given by the same matrices relative to all three ordered bases. This yields 


c_12*(x_0*x_3)+c_13*(-x_1*x_4)+c_14*(-i*x_2*x_4),
c_12*(x_0*x_4)+c_13*(x_1*x_3)+c_14*(-i*x_2*x_3),
c_12*(x_0*x_5)+c_13*(-i*x_2*x_6)+c_14*(-x_1*x_6),
c_12*(x_0*x_6)+c_13*(-i*x_2*x_5)+c_14*(x_1*x_5),
Assume that \( c_{12} \) and \(c_{13}\) are nonzero. Then after scaling \(x_1,x_2\) and dividing, we may assume that \(c_{12} = c_{13} = 1\).

We collect all the polynomials listed above and the assumptions we made about the coefficients \(c_i\) to obtain the following candidate polynomials: \[ \begin{array}{l} x_0^2+ c_3 (x_3 x_4 - i x_5 x_6),\\ x_1^2 + x_2^2+c_6 (x_3 x_4 - i x_5 x_6),\\ x_3 x_6 - x_4 x_5,\\ x_1^2 - x_2^2+c_8 (x_3 x_4 + i x_5 x_6)\\ x_0 x_1+x_3^2-x_4^2+c_{11} (x_5^2-x_6^2),\\ x_0 x_2 -x_5^2-x_6^2+c_{11} (x_3^2+x_4^2),\\ x_0 x_3-x_1 x_4+c_{14} (-i x_2 x_4),\\ x_0 x_4+x_1 x_3+c_{14} (-i x_2 x_3),\\ x_0 x_5-i x_2 x_6+c_{14} (-x_1 x_6),\\ x_0 x_6-i x_2 x_5+c_{14} (x_1 x_5), \end{array} \] For generic values of \(c_3,c_6,c_8, c_{11},c_{14}\), the intersection of these 10 quadrics in \(\mathbb{P}^6\) is empty. Here is an example showing this: 

> > c_3:=1;
> c_6:=1;
> c_8:=1;
> c_11:=1;
> c_14:=1;
> K<i>:=CyclotomicField(4);
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[
> x_0^2+ c_3*(x_3*x_4 - i*x_5*x_6),
> x_1^2 + x_2^2+c_6*(x_3*x_4 - i*x_5*x_6),
> x_3*x_6 - x_4*x_5,
> x_1^2 - x_2^2+c_8*(x_3*x_4 + i*x_5*x_6),
> x_0*x_1+x_3^2-x_4^2+c_11*(x_5^2-x_6^2),
> x_0*x_2-x_5^2-x_6^2+c_11*(x_3^2+x_4^2),
> x_0*x_3-x_1*x_4+c_14*(-i*x_2*x_4),
> x_0*x_4+x_1*x_3+c_14*(-i*x_2*x_3),
> x_0*x_5-i*x_2*x_6+c_14*(-x_1*x_6),
> x_0*x_6-i*x_2*x_5+c_14*(x_1*x_5)
> ]);
> Dimension(X);
-1
Therefore next we turn to Macaulay2 to compute part of a flattening stratification. (We switch software packages because, to the best of our knowledge, Magma will not compute Gröbner bases in a polynomial ring over a polynomial ring.)

Flattening stratification in Macaulay2

We compute the degree two and three elements in a Gröbner basis in Macaulay2 for the ideal generated by the candidate polynomials. 
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : loadPackage("Cyclotomic")

o1 = Cyclotomic

o1 : Package

i2 : K=cyclotomicField(4);

i3 : i=K_0;

i4 : S=K[c_3,c_6,c_8,c_11,c_14,Degrees=>{0,0,0,0,0}]

o4 = S

o4 : PolynomialRing

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

i6 : I=ideal({
     x_0^2+ c_3*(x_3*x_4 - i*x_5*x_6),
     x_1^2 + x_2^2+c_6*(x_3*x_4 - i*x_5*x_6),
     x_3*x_6 - x_4*x_5,
     x_1^2 - x_2^2+c_8*(x_3*x_4 + i*x_5*x_6),
     x_0*x_1+x_3^2-x_4^2+c_11*(x_5^2-x_6^2),
     x_0*x_2-x_5^2-x_6^2+c_11*(x_3^2+x_4^2),
     x_0*x_3-x_1*x_4+c_14*(-i*x_2*x_4),
     x_0*x_4+x_1*x_3+c_14*(-i*x_2*x_3),
     x_0*x_5-i*x_2*x_6+c_14*(-x_1*x_6),
     x_0*x_6-i*x_2*x_5+c_14*(x_1*x_5)
     });

o6 : Ideal of T

i7 : 
     L=flatten entries gens gb(I,DegreeLimit=>6);

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*c_14^2+c_8*c_14^2+2*c_3+c_6-c_8
c_14-1
c_3-ww_4*c_11-c_14
c_6*c_14^2+c_8*c_14^2+c_6-c_8+2*ww_4*c_11+2*c_14
c_11^2+1
1
c_14^2-1
c_6*c_14^2-c_8*c_14^2+2*c_3+c_6+c_8
c_6*c_11+c_8*c_11-ww_4*c_6*c_14+ww_4*c_8*c_14+4*c_11
c_11*c_14+ww_4*c_3-ww_4
c_11*c_14+(1/2)*ww_4*c_3
c_3*c_14-2*c_14
c_3^2-2*c_3
c_11*c_14+ww_4
This suggests setting \(c_{14} =1\) and \(c_{11}=-i\) . We repeat the calculation with these choices: 
i1 : 
     loadPackage("Cyclotomic");

i2 : K=cyclotomicField(4);

i3 : i=K_0;

i4 : S=K[c_3,c_6,c_8,Degrees=>{0,0,0}];

i5 : c_11=-i;

i6 : c_14=1;

i7 : T=S[x_0..x_6];

i8 : I=ideal({
     x_0^2+ c_3*(x_3*x_4 - i*x_5*x_6),
     x_1^2 + x_2^2+c_6*(x_3*x_4 - i*x_5*x_6),
     x_3*x_6 - x_4*x_5,
     x_1^2 - x_2^2+c_8*(x_3*x_4 + i*x_5*x_6),
     x_0*x_1+x_3^2-x_4^2+c_11*(x_5^2-x_6^2),
     x_0*x_2-x_5^2-x_6^2+c_11*(x_3^2+x_4^2),
     x_0*x_3-x_1*x_4+c_14*(-i*x_2*x_4),
     x_0*x_4+x_1*x_3+c_14*(-i*x_2*x_3),
     x_0*x_5-i*x_2*x_6+c_14*(-x_1*x_6),
     x_0*x_6-i*x_2*x_5+c_14*(x_1*x_5)
     });

o8 : Ideal of T

i9 : L=flatten entries gens gb(I,DegreeLimit=>6);

i10 : L3=select(L, i -> degree i == {3,0});

i11 : L3c=unique apply(L3, i -> leadCoefficient i);

i12 : for i from 0 to #L3c-1 do (print toString(L3c_i) << endl)
c_6+2
c_3-2
1
This suggests setting \( c_3 = 2\) and \( c_6=-2.\)

This leads to the following conjectural description of the desired family: for generic values of \(c_8\), \[ \begin{array}{l} x_0^2+ 2 x_3 x_4 - 2 i x_5 x_6,\\ x_1^2 + x_2^2-2 x_3 x_4 +2 i x_5 x_6,\\ x_3 x_6 - x_4 x_5,\\ x_1^2 - x_2^2+c_8 (x_3 x_4 + i x_5 x_6)\\ x_0 x_1+x_3^2-x_4^2-i(x_5^2-x_6^2),\\ x_0 x_2 -x_5^2-x_6^2-i(x_3^2+x_4^2),\\ x_0 x_3-x_1 x_4 -i x_2 x_4,\\ x_0 x_4+x_1 x_3 -i x_2 x_3,\\ x_0 x_5-i x_2 x_6 -x_1 x_6,\\ x_0 x_6-i x_2 x_5+x_1 x_5. \end{array} \] In the next section we show that at least two different values of \(c_{8}\) yield equations of a smooth curve with the correct automorphisms.

Checking the equations in Magma

We check that for two different values of \(c_{8}\), we obtain a a smooth curve with the correct automorphisms. This implies that a general member of the family is a smooth curve with the correct automorphisms. However, I do not know how to show 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 check the value \(c_{8}=1\): 

> K<i>:=CyclotomicField(4);
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c_8:=1;
> X:=Scheme(P6,[
> x_0^2+ 2*x_3*x_4 - 2*i*x_5*x_6,
> x_1^2 + x_2^2-2*x_3*x_4 +2*i*x_5*x_6,
> x_3*x_6 - x_4*x_5,
> x_1^2 - x_2^2+c_8*(x_3*x_4 + i*x_5*x_6),
> x_0*x_1+x_3^2-x_4^2-i*x_5^2+i*x_6^2,
> x_0*x_2-x_5^2-x_6^2-i*x_3^2-i*x_4^2,
> x_0*x_3-x_1*x_4-i*x_2*x_4,
> x_0*x_4+x_1*x_3-i*x_2*x_3,
> x_0*x_5-i*x_2*x_6-x_1*x_6,
> x_0*x_6-i*x_2*x_5+x_1*x_5
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> A:=Matrix([
> [-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]
> ]);
> B:=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, 0, 0, -i],
> [0, 0, 0, 0, 0, -1, 0],
> [0, 0, 0, 0, -1, 0, 0],
> [0, 0, 0, i, 0, 0, 0]
> ]);
> C:=Matrix([
> [-1, 0, 0, 0, 0, 0, 0],
> [0, -1, 0, 0, 0, 0, 0],
> [0, 0, 1, 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]
> ]);
> D:=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, 0, i, 0],
> [0, 0, 0, 0, 0, 0, 1],
> [0, 0, 0, -1, 0, 0, 0],
> [0, 0, 0, 0, i, 0, 0]
> ]);
> GL7K:=GeneralLinearGroup(7,K);
> IdentifyGroup(sub<GL7K | A,B,C>);
<32, 43>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
-x_2
-x_3
-x_4
x_5
x_6
and inverse
-x_0
-x_1
-x_2
-x_3
-x_4
x_5
x_6
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_2
-x_1
i*x_6
-x_5
-x_4
-i*x_3
and inverse
-x_0
-x_2
-x_1
i*x_6
-x_5
-x_4
-i*x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
x_2
i*x_4
-i*x_3
i*x_6
-i*x_5
and inverse
-x_0
-x_1
x_2
i*x_4
-i*x_3
i*x_6
-i*x_5
Next, we check the value \(c_{8}=19-\zeta_{28}^{11}\): 
> K<z_28>:=CyclotomicField(28);
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c_8:=19-z_28^11;
> X:=Scheme(P6,[
> x_0^2+ 2*x_3*x_4 - 2*i*x_5*x_6,
> x_1^2 + x_2^2-2*x_3*x_4 +2*i*x_5*x_6,
> x_3*x_6 - x_4*x_5,
> x_1^2 - x_2^2+c_8*(x_3*x_4 + i*x_5*x_6),
> x_0*x_1+x_3^2-x_4^2-i*x_5^2+i*x_6^2,
> x_0*x_2-x_5^2-x_6^2-i*x_3^2-i*x_4^2,
> x_0*x_3-x_1*x_4-i*x_2*x_4,
> x_0*x_4+x_1*x_3-i*x_2*x_3,
> x_0*x_5-i*x_2*x_6-x_1*x_6,
> x_0*x_6-i*x_2*x_5+x_1*x_5
> ]);
> 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_1
-x_2
-x_3
-x_4
x_5
x_6
and inverse
-x_0
-x_1
-x_2
-x_3
-x_4
x_5
x_6
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_2
-x_1
z_28^7*x_6
-x_5
-x_4
-z_28^7*x_3
and inverse
-x_0
-x_2
-x_1
z_28^7*x_6
-x_5
-x_4
-z_28^7*x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
x_2
z_28^7*x_4
-z_28^7*x_3
z_28^7*x_6
-z_28^7*x_5
and inverse
-x_0
-x_1
x_2
z_28^7*x_4
-z_28^7*x_3
z_28^7*x_6
-z_28^7*x_5

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(4);

i3 : i=K_0;

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

i5 : c_8=1;

i6 : I=ideal {
     x_0^2+ 2*x_3*x_4 - 2*i*x_5*x_6,
     x_1^2 + x_2^2-2*x_3*x_4 +2*i*x_5*x_6,
     x_3*x_6 - x_4*x_5,
     x_1^2 - x_2^2+c_8*(x_3*x_4 + i*x_5*x_6),
     x_0*x_1+x_3^2-x_4^2-i*x_5^2+i*x_6^2,
     x_0*x_2-x_5^2-x_6^2-i*x_3^2-i*x_4^2,
     x_0*x_3-x_1*x_4-i*x_2*x_4,
     x_0*x_4+x_1*x_3-i*x_2*x_3,
     x_0*x_5-i*x_2*x_6-x_1*x_6,
     x_0*x_6-i*x_2*x_5+x_1*x_5
     };

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
By [Schreyer1986], this Betti table implies that this curve has a \(g_6^2\). We get the same Betti table with \(c_{8} = 19-\zeta_{28}^{11}\).