Fordham University

Finding equations of a genus 6 Riemann surface with 48 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 genus 6 Riemann surface with automorphism group (48,15) in the GAP library of small groups. Its quotient surface has genus zero, and the quotient morphism is branched over three points with ramification indices (2,6,8).

We use GAP, Macaulay2 and Magma to compute equations of this Riemann surface.

Obtaining candidate polynomials in Magma and Gap

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     Sat Apr 30 2016 18:59:47 on Davids-MBP-2 [Seed = 1394687225]

+-------------------------------------------------------------------+
|       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"
> G:=SmallGroup(48,15);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,6,[2,6,8]);
Set seed to 0.


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


----------------------------------------------------
Class |   1  2  3  4  5  6  7     8     9  10  11 12
Size  |   1  1  4 12  2  2  2     4     4   6   6  4
Order |   1  2  2  2  3  4  6     6     6   8   8 12
----------------------------------------------------
p  =  2   1  1  1  1  5  2  5     5     5   6   6  7
p  =  3   1  2  3  4  1  6  2     3     3  11  10  6
----------------------------------------------------
X.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
X.3   +   1  1  1 -1  1  1  1     1     1  -1  -1  1
X.4   +   1  1 -1  1  1  1  1    -1    -1  -1  -1  1
X.5   +   2  2  2  0 -1  2 -1    -1    -1   0   0 -1
X.6   +   2  2 -2  0 -1  2 -1     1     1   0   0 -1
X.7   +   2  2  0  0  2 -2  2     0     0   0   0 -2
X.8   0   2  2  0  0 -1 -2 -1-1-2*J 1+2*J   0   0  1
X.9   0   2  2  0  0 -1 -2 -1 1+2*J-1-2*J   0   0  1
X.10  +   2 -2  0  0  2  0 -2     0     0  Z1 -Z1  0
X.11  +   2 -2  0  0  2  0 -2     0     0 -Z1  Z1  0
X.12  +   4 -4  0  0 -2  0  2     0     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)

Z1     = (CyclotomicField(8: 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.4

[3]     Order 2       Length 4      
        Rep G.2

[4]     Order 2       Length 12     
        Rep G.1

[5]     Order 3       Length 2      
        Rep G.5

[6]     Order 4       Length 2      
        Rep G.3

[7]     Order 6       Length 2      
        Rep G.4 * G.5

[8]     Order 6       Length 4      
        Rep G.2 * G.5

[9]     Order 6       Length 4      
        Rep G.2 * G.5^2

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

[11]    Order 8       Length 6      
        Rep G.1 * G.2 * G.3 * G.4

[12]    Order 12      Length 4      
        Rep G.3 * G.5


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

RunExample(
    G: GrpPC : G,
    genus: 6,
    E: [ 2, 6, 8 ]
)
FindMatrixGenerators(
    G: GrpPC : G,
    genus: 6,
    T:   Character Table of Group G --------------------------   --...,
    CCL: Conjugacy Classes of group G ---------------------------- [1...,
    M: [ G.1 * G.3 * G.5^2, G.2 * G.4 * G.5^2, G.1 * G.2 * G.5 ]
)
In file "autcv10e.txt", line 221, column 28:
>>       ags:=ActionGenerators(GModule(T[i]));
                              ^
Runtime error in 'ActionGenerators': Bad argument types
Argument types given: BoolElt

Note that before the error occurred, we learned that this Riemann surface is cyclic trigonal.

The error "No subgroup found" tells us that Magma has an internal error when finding the matrix generators of representations with character \( \chi_{8}\) and/or \( \chi_{12}\).

We obtain these matrices using GAP. First, we check that GAP and Magma use the same polycyclic generators for the group (48,15), 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(48,15));
[ 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 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -1, 0 ], [ 0, -1 ] ], 
      [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], 
      [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -1, 0 ], [ 0, 1 ] ], 
      [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 0, E(8)^3 ], [ -E(8), 0 ] ], [ [ 0, 1 ], [ 1, 0 ] ], 
      [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 0, -E(8)^3 ], [ E(8), 0 ] ], [ [ 0, 1 ], [ 1, 0 ] ], 
      [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -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 ] ], 
      [ [ 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 ] ], 
      [ [ E(3), 0, 0, 0 ], [ 0, E(3), 0, 0 ], [ 0, 0, E(3)^2, 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<z_24>:=CyclotomicField(24);
> z_8:=z_24^3;
> z_4:=z_24^6;
> z_3:=z_24^8;
> 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 ] ] ], 
> [ [ [ 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 ] ], [ [ z_3, 0 ], [ 0, z_3^2 ] ] ], 
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], \
[ [ 1, 0 ], [ 0, 1 ] ], [ [ z_3, 0 ], [ 0, z_3^2 ] ] ], 
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], [ [ -1, 0 ], [ 0, -1 ] ],\
 [ [ 1, 0 ], [ 0, 1 ] ], [ [ z_3, 0 ], [ 0, z_3^2 ] ] ], 
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -1, 0 ], [ 0, 1 ] ], [ [ -1, 0 ], [ 0, -1 ] ],\
 [ [ 1, 0 ], [ 0, 1 ] ], [ [ z_3, 0 ], [ 0, z_3^2 ] ] ], 
> [ [ [ 0, z_8^3 ], [ -z_8, 0 ] ], [ [ 0, 1 ], [ 1, 0 ] ], [ [ z_4, 0 ], [ 0, \
-z_4 ] ], [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], 
> [ [ [ 0, -z_8^3 ], [ z_8, 0 ] ], [ [ 0, 1 ], [ 1, 0 ] ], [ [ z_4, 0 ], [ 0, \
-z_4 ] ], [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], 
> [ [ [ 0, 0, 0, 1 ], [ 0, 0, z_4, 0 ], [ 0, -z_4, 0, 0 ], [ 1, 0, 0, 0 ] ], [\
 [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ], [ [ z_4, 0,\
 0, 0 ], [ 0, -z_4, 0, 0 ], [ 0, 0, z_4, 0 ], [ 0, 0, 0, -z_4 ] ], [ [ -1, 0, \
0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ], [ [ z_3, 0, 0, 0 \
], [ 0, z_3, 0, 0 ], [ 0, 0, z_3^2, 0 ], [ 0, 0, 0, z_3^2 ] ] ]
> ];
> G:=SmallGroup(48,15);
> 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, 3, 4, 2, 7, 5, 6, 9, 8, 10, 11, 12 ]

This tells us that the ninth and twelfth representations in GAP have characters \( \chi_8\) and \( \chi_{12}\) with respect to the Magma character table. Thus, the matrix generators for the representation \( \chi_8 + \chi_{12}\) are: 

> MG:=[DiagonalJoin(Matrix(L[9][i]),Matrix(L[12][i])): i in [1..5]];
> MG;
[
    [      0       1       0       0       0       0]
    [      1       0       0       0       0       0]
    [      0       0       0       0       0       1]
    [      0       0       0       0  z_24^6       0]
    [      0       0       0 -z_24^6       0       0]
    [      0       0       1       0       0       0],

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

    [     -1       0       0       0       0       0]
    [      0      -1       0       0       0       0]
    [      0       0  z_24^6       0       0       0]
    [      0       0       0 -z_24^6       0       0]
    [      0       0       0       0  z_24^6       0]
    [      0       0       0       0       0 -z_24^6],

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

    [z_24^4 - 1          0          0          0          0          0]
    [         0    -z_24^4          0          0          0          0]
    [         0          0 z_24^4 - 1          0          0          0]
    [         0          0          0 z_24^4 - 1          0          0]
    [         0          0          0          0    -z_24^4          0]
    [         0          0          0          0          0    -z_24^4]
]
> A:=MG[1]*MG[3]*MG[5]^2;
> B:=MG[2]*MG[4]*MG[5]^2;
> A;
[0 -z_24^4 + 1 0 0 0 0]
[z_24^4 0 0 0 0 0]
[0 0 0 0 0 z_24^2]
[0 0 0 0 -z_24^4 + 1 0]
[0 0 0 z_24^4 0 0]
[0 0 -z_24^6 + z_24^2 0 0 0]
> B;
[     z_24^4           0           0           0           0           0]
[          0  z_24^4 - 1           0           0           0           0]
[          0           0           0      z_24^4           0           0]
[          0           0      z_24^4           0           0           0]
[          0           0           0           0           0 -z_24^4 + 1]
[          0           0           0           0 -z_24^4 + 1           0]

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

> MatrixG,MatrixSKG,Q,C:=RunGivenSKMatrixGenerators(48,6,[A,B,(A*B)^-1]);
Set seed to 0.


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


----------------------------------------------------
Class |   1  2  3  4  5  6  7     8     9  10  11 12
Size  |   1  1  4 12  2  2  2     4     4   6   6  4
Order |   1  2  2  2  3  4  6     6     6   8   8 12
----------------------------------------------------
p  =  2   1  1  1  1  5  2  5     5     5   6   6  7
p  =  3   1  2  3  4  1  6  2     3     3  11  10  6
----------------------------------------------------
X.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
X.3   +   1  1  1 -1  1  1  1     1     1  -1  -1  1
X.4   +   1  1 -1  1  1  1  1    -1    -1  -1  -1  1
X.5   +   2  2  2  0 -1  2 -1    -1    -1   0   0 -1
X.6   +   2  2 -2  0 -1  2 -1     1     1   0   0 -1
X.7   +   2  2  0  0  2 -2  2     0     0   0   0 -2
X.8   0   2  2  0  0 -1 -2 -1-1-2*J 1+2*J   0   0  1
X.9   0   2  2  0  0 -1 -2 -1 1+2*J-1-2*J   0   0  1
X.10  +   2 -2  0  0  2  0 -2     0     0  Z1 -Z1  0
X.11  +   2 -2  0  0  2  0 -2     0     0 -Z1  Z1  0
X.12  +   4 -4  0  0 -2  0  2     0     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)

Z1     = (CyclotomicField(8: Sparse := true)) ! [ RationalField() | 0, -1, 0, 1 
]


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

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

[3]     Order 2       Length 4      
        Rep [-1  0  0  0  0  0]
        [ 0  1  0  0  0  0]
        [ 0  0  0 -1  0  0]
        [ 0  0 -1  0  0  0]
        [ 0  0  0  0  0 -1]
        [ 0  0  0  0 -1  0]

[4]     Order 2       Length 12     
        Rep [          0    -z^8 + 1           0           0           0        
        0]
        [        z^8           0           0           0           0           
        0]
        [          0           0           0           0           0         
            z^4]
        [          0           0           0           0    -z^8 + 1           
        0]
        [          0           0           0         z^8           0           
        0]
        [          0           0 -z^12 + z^4           0           0           
        0]

[5]     Order 3       Length 2      
        Rep [z^8 - 1       0       0       0       0       0]
        [      0    -z^8       0       0       0       0]
        [      0       0 z^8 - 1       0       0       0]
        [      0       0       0 z^8 - 1       0       0]
        [      0       0       0       0    -z^8       0]
        [      0       0       0       0       0    -z^8]

[6]     Order 4       Length 2      
        Rep [   -1     0     0     0     0     0]
        [    0    -1     0     0     0     0]
        [    0     0 -z^12     0     0     0]
        [    0     0     0  z^12     0     0]
        [    0     0     0     0 -z^12     0]
        [    0     0     0     0     0  z^12]

[7]     Order 6       Length 2      
        Rep [ z^8 - 1        0        0        0        0        0]
        [       0     -z^8        0        0        0        0]
        [       0        0 -z^8 + 1        0        0        0]
        [       0        0        0 -z^8 + 1        0        0]
        [       0        0        0        0      z^8        0]
        [       0        0        0        0        0      z^8]

[8]     Order 6       Length 4      
        Rep [     z^8        0        0        0        0        0]
        [       0  z^8 - 1        0        0        0        0]
        [       0        0        0      z^8        0        0]
        [       0        0      z^8        0        0        0]
        [       0        0        0        0        0 -z^8 + 1]
        [       0        0        0        0 -z^8 + 1        0]

[9]     Order 6       Length 4      
        Rep [-z^8 + 1        0        0        0        0        0]
        [       0     -z^8        0        0        0        0]
        [       0        0        0 -z^8 + 1        0        0]
        [       0        0 -z^8 + 1        0        0        0]
        [       0        0        0        0        0      z^8]
        [       0        0        0        0      z^8        0]

[10]    Order 8       Length 6      
        Rep [    0    -1     0     0     0     0]
        [    1     0     0     0     0     0]
        [    0     0     0     0 -z^12     0]
        [    0     0     0     0     0    -1]
        [    0     0    -1     0     0     0]
        [    0     0     0  z^12     0     0]

[11]    Order 8       Length 6      
        Rep [    0     1     0     0     0     0]
        [   -1     0     0     0     0     0]
        [    0     0     0     0     1     0]
        [    0     0     0     0     0  z^12]
        [    0     0 -z^12     0     0     0]
        [    0     0     0     1     0     0]

[12]    Order 12      Length 4      
        Rep [        z^8           0           0           0           0        
        0]
        [          0    -z^8 + 1           0           0           0           
        0]
        [          0           0 -z^12 + z^4           0           0           
        0]
        [          0           0           0  z^12 - z^4           0           
        0]
        [          0           0           0           0        -z^4           
        0]
        [          0           0           0           0           0         
            z^4]


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, 0, 0, 0 ]
S_1      =[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1 ]
I_2      =[ 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0 ]
S_2      =[ 1, 0, 0, 2, 2, 0, 1, 1, 1, 1, 1, 1 ]
H^0(C,2K)=[ 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1 ]
I_3      =[ 1, 0, 0, 0, 1, 1, 1, 2, 1, 2, 1, 3 ]
S_3      =[ 1, 1, 1, 1, 2, 2, 2, 3, 1, 3, 3, 5 ]
H^0(C,3K)=[ 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2 ]
I2timesS1=[ 0, 0, 0, 0, 1, 1, 1, 3, 2, 1, 1, 4 ]
Is clearly not generated by quadrics? true
Plane quintic obstruction?  true
Matrix Surface Kernel Generators:
Field K Cyclotomic Field of order 48 and degree 16
[
    [          0    -z^8 + 1           0           0           0           0]
    [        z^8           0           0           0           0           0]
    [          0           0           0           0           0         z^4]
    [          0           0           0           0    -z^8 + 1           0]
    [          0           0           0         z^8           0           0]
    [          0           0 -z^12 + z^4           0           0           0],

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

    [          0        -z^8           0           0           0           0]
    [   -z^8 + 1           0           0           0           0           0]
    [          0           0           0           0        -z^8           0]
    [          0           0           0           0           0 -z^12 + z^4]
    [          0           0         z^4           0           0           0]
    [          0           0           0     z^8 - 1           0           0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 1
Multiplicity 1
[
    x_2*x_5 + x_3*x_4
]
I2 contains a 1-dimensional subspace of CharacterRow 4
Dimension 2
Multiplicity 2
[
    x_0*x_1,
    x_2*x_5 - x_3*x_4
]
I2 contains a 2-dimensional subspace of CharacterRow 5
Dimension 4
Multiplicity 2
[
    x_0^2,
    x_1^2,
    x_2*x_3,
    x_4*x_5
]
I2 contains a 2-dimensional subspace of CharacterRow 10
Dimension 2
Multiplicity 1
[
    x_0*x_4 + (z^10 - z^2)*x_1*x_2,
    x_0*x_5 + (-z^10 + z^2)*x_1*x_3
]
Finding cubics:
I3 contains a 1-dimensional subspace of CharacterRow 1
Dimension 1
Multiplicity 1
[
    x_0*x_2^2 - x_0*x_3^2 + x_1*x_4^2 + x_1*x_5^2
]
I3 contains a 2-dimensional subspace of CharacterRow 5
Dimension 4
Multiplicity 2
[
    x_0*x_2*x_4 - x_0*x_3*x_5,
    x_0*x_4^2 - x_0*x_5^2,
    x_1*x_2^2 + x_1*x_3^2,
    x_1*x_2*x_4 + x_1*x_3*x_5
]
I3 contains a 2-dimensional subspace of CharacterRow 6
Dimension 4
Multiplicity 2
[
    x_0*x_2*x_4 + x_0*x_3*x_5,
    x_0*x_4^2 + x_0*x_5^2,
    x_1*x_2^2 - x_1*x_3^2,
    x_1*x_2*x_4 - x_1*x_3*x_5
]
I3 contains a 2-dimensional subspace of CharacterRow 7
Dimension 4
Multiplicity 2
[
    x_0^3,
    x_0*x_2*x_3,
    x_1^3,
    x_1*x_4*x_5
]
I3 contains a 4-dimensional subspace of CharacterRow 8
Dimension 6
Multiplicity 3
[
    x_0^2*x_1,
    x_0*x_1^2,
    x_0*x_2*x_5 - x_0*x_3*x_4,
    x_0*x_4*x_5,
    x_1*x_2*x_3,
    x_1*x_2*x_5 - x_1*x_3*x_4
]
I3 contains a 2-dimensional subspace of CharacterRow 9
Dimension 2
Multiplicity 1
[
    x_0*x_2*x_5 + x_0*x_3*x_4,
    x_1*x_2*x_5 + x_1*x_3*x_4
]
I3 contains a 4-dimensional subspace of CharacterRow 10
Dimension 6
Multiplicity 3
[
    x_0^2*x_2 + (-z^10 + z^2)*x_1^2*x_4,
    x_0^2*x_3 + (-z^10 + z^2)*x_1^2*x_5,
    x_2^3 + z^6*x_4^3,
    x_2^2*x_3 - z^6*x_4^2*x_5,
    x_2*x_3^2 - z^6*x_4*x_5^2,
    x_3^3 + z^6*x_5^3
]
I3 contains a 2-dimensional subspace of CharacterRow 11
Dimension 6
Multiplicity 3
[
    x_0^2*x_2 + (z^10 - z^2)*x_1^2*x_4,
    x_0^2*x_3 + (z^10 - z^2)*x_1^2*x_5,
    x_2^3 - z^6*x_4^3,
    x_2^2*x_3 + z^6*x_4^2*x_5,
    x_2*x_3^2 + z^6*x_4*x_5^2,
    x_3^3 - z^6*x_5^3
]
I3 contains a 12-dimensional subspace of CharacterRow 12
Dimension 20
Multiplicity 5
[
    x_0^2*x_4,
    x_0^2*x_5,
    x_0*x_1*x_2,
    x_0*x_1*x_3,
    x_0*x_1*x_4,
    x_0*x_1*x_5,
    x_1^2*x_2,
    x_1^2*x_3,
    x_2^2*x_4,
    x_2^2*x_5,
    x_2*x_3*x_4,
    x_2*x_3*x_5,
    x_2*x_4^2,
    x_2*x_4*x_5,
    x_2*x_5^2,
    x_3^2*x_4,
    x_3^2*x_5,
    x_3*x_4^2,
    x_3*x_4*x_5,
    x_3*x_5^2
]

This Riemann surface is cyclic trigonal, and hence the quadrics in its canonical ideal cut out a scroll. We begin by finding quadrics that define a smooth surface.

The first isotypical subspace in \(S_2\), which corresponds to the character \( \chi_1\) in the character table shown above, yields the polynomial 


x_2*x_5 + x_3*x_4

The second isotypical subspace in \(S_2\), which corresponds to the character \( \chi_4\) in the character table shown above, yields a polynomial of the form 


c_1*(x_0*x_1) + c_2*(x_2*x_5 - x_3*x_4)
Assume that \(c_1\) is nonzero. Then after dividing by \(c_1\), we may assume \(c_1=1\).

The third isotypical subspace in \(S_2\), which corresponds to the character \( \chi_5\) in the character table shown above, yields the polynomial the form 


x_3*x_6 - x_4*x_5.

The third isotypical subspace corresponds to the character \( \chi_{5}\) in the character table shown above. Note that the matrix surface kernel generators have a block diagonal form with blocks of size \(2 \times 2\) and \(4 \times 4\). We therefore let the first two polynomials shown here generate one copy of \(V_{5}\) and find an ordered basis of the span of the last two polynomials such that the action of \(G\) is given by the same matrices relative to both bases. 

> K<z_48>:=CyclotomicField(48);
> GL6K:=GeneralLinearGroup(6,K);
> MatrixG:=sub<GL6K | MatrixSKG>;
> FindParallelBases(MatrixG,[Q[3][1],Q[3][2]],[Q[3][3],Q[3][4]]);
[         x_2*x_3]
[-z_48^12*x_4*x_5]
This yields the polynomials 

c_3*(x_0^2)+c_4*(x_2*x_3)
c_3*(x_1^2)+c_4*(-z_48^12*x_4*x_5),

Assume that \(c_3,c_4\) are nonzero. After scaling \(x_0,x_1\) and dividing by \(c_3\), we may assume that \(c_3 = c_4= 1.\)

The fourth isotypical subspace corresponds to the character \( \chi_{10}\) in the character table shown above. It yields the polynomials 


x_0*x_4 + (z^10 - z^2)*x_1*x_2,
x_0*x_5 + (-z^10 + z^2)*x_1*x_3

For generic values of \(c_2\), the intersection of these six quadrics in \(\mathbb{P}^5\) is one-dimensional. Here is an example showing this: 
> K<z_48>:=CyclotomicField(48);
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> c_2:=23-z_48^19;
> Y:=Scheme(P5,[
> x_2*x_5 + x_3*x_4,
> x_0*x_1+c_2*(x_2*x_5 - x_3*x_4),
> x_0^2+x_2*x_3,
> x_1^2-z_48^12*x_4*x_5,
> x_0*x_4 + (z_48^10 - z_48^2)*x_1*x_2,
> x_0*x_5 + (-z_48^10 + z_48^2)*x_1*x_3
> ]);
> Dimension(Y);
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.) 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");

i2 : K=cyclotomicField(48);

i3 : z_48=K_0;

i4 : S=K[c_2,Degrees=>{0}]

o4 = S

o4 : PolynomialRing

i5 : T=S[x_0..x_5];

i6 : I=ideal({
     x_2*x_5 + x_3*x_4,
     x_0*x_1+c_2*(x_2*x_5 - x_3*x_4),
     x_0^2+x_2*x_3,
     x_1^2-z_48^12*x_4*x_5,
     x_0*x_4 + (z_48^10 - z_48^2)*x_1*x_2,
     x_0*x_5 + (-z_48^10 + z_48^2)*x_1*x_3
     });

o6 : Ideal of T

i7 : L=flatten entries gens gb(I);

i8 : Lc=unique apply(L, i -> leadCoefficient i);

i9 : for i from 0 to #Lc-1 do (print toString(Lc_i) << endl)
1
c_2+(1/2)*ww_48^6
This suggests setting \(c_2 =-\frac{1}{2}\zeta_8.\) We check in Magma that this yields a smooth surface: 
> K<z_48>:=CyclotomicField(48);
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> c_2:=-(1/2)*z_48^6;
> Y:=Scheme(P5,[
> x_2*x_5 + x_3*x_4,
> x_0*x_1+c_2*(x_2*x_5 - x_3*x_4),
> x_0^2+x_2*x_3,
> x_1^2-z_48^12*x_4*x_5,
> x_0*x_4 + (z_48^10 - z_48^2)*x_1*x_2,
> x_0*x_5 + (-z_48^10 + z_48^2)*x_1*x_3
> ]);
> Dimension(Y);
2
> IsSingular(Y);
false

Next, we seek the cubic generators in the canonical ideal. These come from the isotypical subspaces of \(S_3\) with characters \(\chi_1\) and \(\chi_{10}\).

The isotypical subspace of \(S_3\) with character \(\chi_1\) yields the polynomial 


x_0*x_2^2 - x_0*x_3^2 + x_1*x_4^2 + x_1*x_5^2

The isotypical subspace of \(S_3\) with character \(\chi_{10}\) has multiplicity three. We let one copy of \(V_{10}\) be generated by the first two cubics shown, and find two more ordered bases such that the matrix action on all three bases is the same. 


> FindParallelBases(MatrixG,[C[7][1],C[7][2]],[C[7][3],C[7][6]]);
[x_3^3 + z_48^6*x_5^3]
[x_2^3 + z_48^6*x_4^3]
> FindParallelBases(MatrixG,[C[7][1],C[7][2]],[C[7][4],C[7][5]]);
[x_2^2*x_3 - z_48^6*x_4^2*x_5]
[x_2*x_3^2 - z_48^6*x_4*x_5^2]
This yields the polynomials 

c_5*(x_0^2*x_2 + (-z_48^10 + z_48^2)*x_1^2*x_4) + c_6*(x_3^3 + z_48^6*x_5^3) +c_7*(x_2^2*x_3 - z_48^6*x_4^2*x_5)
c_5*(x_0^2*x_3 + (-z_48^10 + z_48^2)*x_1^2*x_5) + c_6*(x_2^3 + z_48^6*x_4^3) +c_7*(x_2*x_3^2 - z_48^6*x_4*x_5^2)
After subtracting multiples of the quadrics we may assume that \(c_5=0\). Assume that \(c_6\) is nonzero. Then after dividing by \(c_6\) we may assume \(c_6=1\). Again we compute a partial flattening stratification to find a value of \(c_7\) that yields a smooth cuve. 
i1 : loadPackage("Cyclotomic");

i2 : K=cyclotomicField(48);

i3 : z_48=K_0;

i4 : S=K[c_7,Degrees=>{0}];

i5 : c_2=-1/2*z_48^6;

i6 : T=S[x_0..x_5];

i7 : I=ideal({
     x_2*x_5 + x_3*x_4,
     x_0*x_1+c_2*(x_2*x_5 - x_3*x_4),
     x_0^2+x_2*x_3,
     x_1^2-z_48^12*x_4*x_5,
     x_0*x_4 + (z_48^10 - z_48^2)*x_1*x_2,
     x_0*x_5 + (-z_48^10 + z_48^2)*x_1*x_3,
     x_0*x_2^2 - x_0*x_3^2 + x_1*x_4^2 + x_1*x_5^2,
     x_3^3 + z_48^6*x_5^3 +c_7*(x_2^2*x_3 - z_48^6*x_4^2*x_5),
     x_2^3 + z_48^6*x_4^3 +c_7*(x_2*x_3^2 - z_48^6*x_4*x_5^2)
     });

o7 : Ideal of T

i8 : L=flatten entries gens gb(I);

i9 : Lc=unique apply(L, i -> leadCoefficient i);

i10 : for i from 0 to #Lc-1 do (print toString(Lc_i) << endl)
1
c_7
c_7+1
This suggest \(c_7=-1\). We check in Magma that this yields a smooth curve: 
> K<z_48>:=CyclotomicField(48);
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> X:=Scheme(P5,[
> x_2*x_5 + x_3*x_4,
> x_0*x_1 -(1/2)*z_48^6*(x_2*x_5 - x_3*x_4),
> x_0^2+x_2*x_3,
> x_1^2-z_48^12*x_4*x_5,
> x_0*x_4 + (z_48^10 - z_48^2)*x_1*x_2,
> x_0*x_5 + (-z_48^10 + z_48^2)*x_1*x_3,
> x_0*x_2^2 - x_0*x_3^2 + x_1*x_4^2 + x_1*x_5^2,
> x_3^3 + z_48^6*x_5^3 - x_2^2*x_3 + z_48^6*x_4^2*x_5,
> x_2^3 + z_48^6*x_4^3 -x_2*x_3^2 + z_48^6*x_4*x_5^2
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
10*$.1 - 5
2
> z_24:=z_48^2;
> A:=Matrix([
> [0, -z_24^4 + 1, 0, 0, 0, 0],
> [z_24^4, 0, 0, 0, 0, 0],
> [0, 0, 0, 0, 0, z_24^2],
> [0, 0, 0, 0, -z_24^4 + 1, 0],
> [0, 0, 0, z_24^4, 0, 0],
> [0, 0, -z_24^6 + z_24^2, 0, 0, 0]
> ]);
> B:=Matrix([
> [z_24^4, 0, 0, 0, 0, 0],
> [0, z_24^4 - 1, 0, 0, 0, 0],
> [0, 0, 0, z_24^4, 0, 0],
> [0, 0, z_24^4, 0, 0, 0],
> [0, 0, 0, 0, 0, -z_24^4 + 1],
> [0, 0, 0, 0, -z_24^4 + 1, 0]
> ]);
> Order(A);
2
> Order(B);
6
> Order( (A*B)^-1);
8
> GL6K:=GeneralLinearGroup(6,K);
> IdentifyGroup(sub<GL6K | A,B>);
<48, 15>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
z_48^8*x_1
(-z_48^8 + 1)*x_0
(-z_48^12 + z_48^4)*x_5
z_48^8*x_4
(-z_48^8 + 1)*x_3
z_48^4*x_2
and inverse
z_48^8*x_1
(-z_48^8 + 1)*x_0
(-z_48^12 + z_48^4)*x_5
z_48^8*x_4
(-z_48^8 + 1)*x_3
z_48^4*x_2
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
z_48^8*x_0
(z_48^8 - 1)*x_1
z_48^8*x_3
z_48^8*x_2
(-z_48^8 + 1)*x_5
(-z_48^8 + 1)*x_4
and inverse
(-z_48^8 + 1)*x_0
-z_48^8*x_1
(-z_48^8 + 1)*x_3
(-z_48^8 + 1)*x_2
z_48^8*x_5
z_48^8*x_4

Cyclic trigonal equation

We find a change of variables to transform the quadrics into standard scroll equations. From the list of conjugacy classes above, we see that the only order three elements belong to the fifth conjugacy class, so these must be the cyclic trigonal morphisms. The eigenvalues of the representative shown are \( \zeta_3 \) with multiplicity 3 and \( \zeta_3^2 \) with multiplicity 3. Thus, in the notation of [AchterPries2007] we have \(r = 3, s=3\), so \(d_1 = 4, d_2 = 4\), and the target scroll is given by the \(2 \times 2\) minors of \[ \left[ \begin{array}{rrrrr} x_0 & x_1 & x_3 & x_4 \\ x_1 & x_2 & x_4 & x_5 \end{array} \right]. \] We make a series of transformations to map the quadrics above to these quadrics. First, scaling \(x_1\) by \(\zeta_8\) and \(x_3\) by \(-1\) yields the ideal \[ \begin{array}{c} -x_3 x_4+x_2 x_5,\\ x_0 x_1-x_2 x_5,\\ x_0^2-x_2 x_3,\\ x_1^2-x_4 x_5,\\ -x_1 x_2+x_0 x_4,\\ -x_1 x_3+x_0 x_5 \end{array} \] Then permuting the variables by \[ \begin{array}{rcl} x_0 & \mapsto & x_4 \\ x_2 & \mapsto & x_5 \\ x_4 & \mapsto & x_2 \\ x_5 & \mapsto & x_0 \end{array} \] yields the desired ideal of quadrics. These coordinate transformations map the canonical ideal to the ideal generated by the scroll quadrics and \[ \begin{array}{c} \zeta_8 x_0^2 x_1+\zeta_8 x_1 x_2^2-x_3^2 x_4+x_4 x_5^2,\\ \zeta_8 x_0^3+\zeta_8 x_0 x_2^2+x_3^3-x_3 x_5^2,\\ \zeta_8 x_0^2 x_2+\zeta_8 x_2^3-x_3^2 x_5+x_5^3 \end{array} \] Scaling \(x_0,x_1,x_2\) by \(zeta_{24}^{-1}\) yields the cubics \[ \begin{array}{c} x_0^2 x_1+x_1 x_2^2-x_3^2 x_4+x_4 x_5^2\\ x_0^3+x_0 x_2^2+x_3^3-x_3 x_5^2\\ x_0^2 x_2+x_2^3-x_3^2 x_5+x_5^3 \end{array} \] These encode the cyclic trigonal equation \( y^3 = (x^4-1)^2(x^4+1)\). We transform the matrix surface kernel generators using the maps described above and check everything one last time in Magma.

Checking the equations in Magma

> K<z_24>:=CyclotomicField(24);
> z_8:=z_24^3;
> i:=z_24^6;
> z_6:=z_24^4;
> z_3:=z_24^8;
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> X:=Scheme(P5,[
> -x_4^2 + x_3*x_5,
> -x_2*x_4 + x_1*x_5,
> -x_2*x_3 + x_1*x_4,
> -x_2*x_3 + x_0*x_5,
> -x_1*x_3 + x_0*x_4,
> -x_1^2 + x_0*x_2,
> x_0*x_1^2 - x_2^3 - x_3*x_4^2 - x_5^3,
> x_0^2*x_1 - x_1*x_2^2 - x_3^2*x_4 - x_4*x_5^2,
> x_0^3 - x_1^2*x_2 - x_3^3 - x_4^2*x_5
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
10*$.1 - 5
2
> A:=Matrix([
> [0,0,0,0,0,-z_8],
> [0,0,0,0,i,0],
> [0,0,0,-z_8^3,0,0],
> [0,0,z_8,0,0,0],
> [0,-i,0,0,0,0],
> [z_8^3,0,0,0,0,0]
> ]);
> B:=Matrix([
> [0,0,-z_6,0,0,0],
> [0,z_6,0,0,0,0],
> [-z_6,0,0,0,0,0],
> [0,0,0,0,0,-z_3],
> [0,0,0,0,z_3,0],
> [0,0,0,-z_3,0,0]
> ]);
> Order(A);
2
> Order(B);
6
> Order( (A*B)^(-1));
8
> GL6K:=GeneralLinearGroup(6,K);
> IdentifyGroup(sub<GL6K | A,B>);
<48, 15>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
(z_24^5 - z_24)*x_5
-z_24^6*x_4
z_24^3*x_3
(-z_24^5 + z_24)*x_2
z_24^6*x_1
-z_24^3*x_0
and inverse
(z_24^5 - z_24)*x_5
-z_24^6*x_4
z_24^3*x_3
(-z_24^5 + z_24)*x_2
z_24^6*x_1
-z_24^3*x_0
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-z_24^4*x_2
z_24^4*x_1
-z_24^4*x_0
(-z_24^4 + 1)*x_5
(z_24^4 - 1)*x_4
(-z_24^4 + 1)*x_3
and inverse
(z_24^4 - 1)*x_2
(-z_24^4 + 1)*x_1
(z_24^4 - 1)*x_0
z_24^4*x_5
-z_24^4*x_4
z_24^4*x_3