|
Magaard, Shaska, Shpectorov, and Völklein give tables of smooth Riemann surfaces of genus \( g \leq 10\) with automorphism groups \(G\) satisfying \( \# G > 4(g-1)\). Their list is based on a computer search by Breuer.
They list a genus 5 Riemann surface with automorphism group (64,32) in the GAP library of small groups. The quotient of this surface by its automorphism group has genus zero, and the quotient morphism is branched over three points with ramification indices (2,4,8).
We use GAP and Magma to compute equations of this Riemann surface. The main tools are the Eichler trace formula, black-box commands in GAP and Magma for obtaining matrix generators of a representation of a finite group having a specified character.
Magma V2.21-4 Sun Sep 13 2015 09:44:36 on ace-math01 [Seed = 730530581]
Type ? for help. Type -D to quit.
> load "autcv10.txt";
Loading "autcv10.txt"
> G:=SmallGroup(64,32);
> G;
GrpPC : G of order 64 = 2^6
PC-Relations:
G.1^2 = G.4,
G.2^G.1 = G.2 * G.3,
G.3^G.1 = G.3 * G.5,
G.4^G.2 = G.4 * G.5,
G.4^G.3 = G.4 * G.6,
G.5^G.1 = G.5 * G.6
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,5,[2,4,8]);
Character Table of Group G
--------------------------
-----------------------------------------------
Class | 1 2 3 4 5 6 7 8 9 10 11 12 13
Size | 1 1 2 4 4 4 4 4 8 8 8 8 8
Order | 1 2 2 2 2 2 2 4 4 4 4 8 8
-----------------------------------------------
p = 2 1 1 1 1 1 1 1 2 6 3 6 8 8
-----------------------------------------------
X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 + 1 1 1 -1 -1 1 1 1 1 -1 1 -1 -1
X.3 + 1 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 1
X.5 0 1 1 1 1 1 -1 1 -1 I -1 -I I -I
X.6 0 1 1 1 -1 -1 -1 1 -1 I 1 -I -I I
X.7 0 1 1 1 1 1 -1 1 -1 -I -1 I -I I
X.8 0 1 1 1 -1 -1 -1 1 -1 -I 1 I I -I
X.9 + 2 2 2 0 0 2 -2 -2 0 0 0 0 0
X.10 + 2 2 2 0 0 -2 -2 2 0 0 0 0 0
X.11 + 4 -4 0 2 -2 0 0 0 0 0 0 0 0
X.12 + 4 -4 0 -2 2 0 0 0 0 0 0 0 0
X.13 + 4 4 -4 0 0 0 0 0 0 0 0 0 0
Explanation of Character Value Symbols
--------------------------------------
I = RootOfUnity(4)
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 1
Rep G.6
[3] Order 2 Length 2
Rep G.5
[4] Order 2 Length 4
Rep G.2 * G.6
[5] Order 2 Length 4
Rep G.2
[6] Order 2 Length 4
Rep G.4
[7] Order 2 Length 4
Rep G.3
[8] Order 4 Length 4
Rep G.3 * G.4
[9] Order 4 Length 8
Rep G.1
[10] Order 4 Length 8
Rep G.2 * G.4
[11] Order 4 Length 8
Rep G.1 * G.4
[12] Order 8 Length 8
Rep G.1 * G.2
[13] Order 8 Length 8
Rep G.1 * G.2 * G.4
SKGs: [ G.2 * G.5 * G.6, G.1, G.1 * G.2 * G.4 * G.6 ]
Is hyperelliptic? false
Is cyclic trigonal? false
Multiplicities of irreducibles in relevant G-modules:
I_1 =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1 =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0 ]
H^0(C,1K)=[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0 ]
I_2 =[ 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_2 =[ 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ]
H^0(C,2K)=[ 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1 ]
I_3 =[ 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0 ]
S_3 =[ 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 5, 1 ]
H^0(C,3K)=[ 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1 ]
I2timesS1=[ 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0 ]
Is clearly not generated by quadrics? false
No subgroup found
RunExample(
G: GrpPC : G,
genus: 5,
E: [ 2, 4, 8 ]
)
findMatrixGenerators(
G: GrpPC : G,
genus: 5,
T: Character Table of Group G -------------------------- --...,
CCL: Conjugacy Classes of group G ---------------------------- [1...,
M: [ G.2 * G.5 * G.6, G.1, G.1 * G.2 * G.4 * G.6 ]
)
In file "autcv8.txt", line 327, column 22:
>> 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 \( \chi_{12}\) and/or \( \chi_{6}\).
To find these matrix generators, we swtich to using the GAP computational algebra system. We first check that GAP and Magma use the same generators for the group (64,32) by checking that the GAP generators satisfy the Magma relations.
gap> G:=SmallGroup(64,32);
gap> (G.2^G.1)*(G.2 * G.3)^-1;
<identity> of ...
gap> (G.3^G.1)*(G.3 * G.5)^-1;
<identity> of ...
gap> (G.4^G.2)*(G.4 * G.5)^-1;
<identity> of ...
gap> (G.4^G.3)*(G.4 * G.6)^-1;
<identity> of ...
gap> (G.5^G.1)*(G.5 * G.6)^-1;
<identity> of ...
gap> IrreducibleRepresentations(G);
[ Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ],
[ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
[ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ],
[ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
[ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ [ [ E(4) ] ], [ [ 1 ] ], [ [ 1 ] ],
[ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
[ [ [ -E(4) ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ [ [ E(4) ] ], [ [ -1 ] ], [ [ 1 ] ],
[ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
[ [ [ -E(4) ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ]
, Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
[ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ],
[ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ],
[ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ],
Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
[ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ],
[ [ -1, 0 ], [ 0, -1 ] ], [ [ -1, 0 ], [ 0, -1 ] ],
[ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ],
Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
[ [ [ 0, 0, 1, 0 ], [ 0, 0, 0, -1 ], [ 1, 0, 0, 0 ], [ 0, 1, 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 ] ],
[ [ 1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, -1 ] ],
[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ],
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ] ],
Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
[ [ [ 0, 0, 1, 0 ], [ 0, 0, 0, -1 ], [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ] ],
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 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 ] ],
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ],
[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ] ]
, Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
[ [ [ 0, 0, 1, 0 ], [ 0, 0, 0, -1 ], [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ] ],
[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, -1, 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 ] ],
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ],
[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ]
] ]
Next, 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(4);
> 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 ] ] ]
> [ [ [ (z) ] ], [ [ 1 ] ], [ [ 1 ] ],[ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ (z) ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ (z) ] ], [ [ -1 ] ], [ [ 1 ] ],[ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ -(z) ] ], [ [ -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 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ],
> [ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ],[ [ -1, 0 ], [ 0, -1 ] ][ [ -1,
0 ], [ 0, -1 ] ],[ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ],
>[ [ [ 0, 0, 1, 0 ], [ 0, 0, 0, -1 ], [ 1, 0, 0, 0 ], [ 0, 1, 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 ] ],[ [ 1, 0, 0, 0 ], [ 0, -1,0,0 ], [ 0, 0,
1, 0 ], [ 0, 0, 0, -1 ] ],[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ],[ 0, 0, -1, 0 ],[ 0,
0,0, -1 ] ],[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0,1, 0], [ 0, 0, 0, 1 ] ] ],
> [ [ [ 0, 0, 1, 0 ], [ 0, 0, 0, -1 ], [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ] ],[ [ 1,0, 0,
0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 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 ] ], [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [0, 0, -1, 0 ], [ 0, 0, 0,
-1 ] ],[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ] ],
> [ [ [ 0, 0, 1, 0 ], [ 0, 0, 0, -1 ], [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ] ],[ [ -1, 0,0,
0 ], [ 0, -1, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, -1, 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 ] ],[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ],[ 0, 0, -1, 0 ], [ 0, 0,
0, -1 ] ],[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0,-1, 0 ], [ 0, 0, 0, -1 ] ]]
> ];
> G:=SmallGroup(64,32);
> 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, 2, 4, 5, 7, 6, 8, 9, 10, 13, 12, 11 ]
This tells us that that seventh and twelfth homomorphisms in this list correspond to the characters \( \chi_6\) and \( \chi_{12}\) in the character table shown above. Thus, the matrix generators for the representation \( \chi_6 + \chi_{12}\) are:
> MG:=[Matrix([[z,0,0,0,0],[0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, -1 ], [ 0, 1, 0, 0,0 ],
[ 0, 0, 1, 0,0 ]]), Matrix([[-1,0,0,0,0],[ 0, 1,0, 0, 0 ], [ 0, 0, 1, 0,0 ], [ 0, 0,
0, 0, 1 ], [ 0, 0, 0, 1,0 ]]), Matrix([[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 ]]), Matrix([[-1,0,0,0,0],[ 0, 1, 0, 0, 0 ],
[ 0, 0, -1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, -1 ] ]), Matrix([[1,0,0,0,0],
[ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [0, 0, 0, -1, 0 ], [ 0, 0, 0, 0,-1 ]]),
Matrix([[1,0,0,0,0],[ 0, -1, 0, 0, 0 ], [ 0, 0, -1, 0, 0 ], [ 0, 0, 0, -1,0 ], [ 0,
0, 0, 0, -1 ]])];
> MG;
[
[ z 0 0 0 0]
[ 0 0 0 1 0]
[ 0 0 0 0 -1]
[ 0 1 0 0 0]
[ 0 0 1 0 0],
[-1 0 0 0 0]
[ 0 1 0 0 0]
[ 0 0 1 0 0]
[ 0 0 0 0 1]
[ 0 0 0 1 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],
[-1 0 0 0 0]
[ 0 1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 1 0]
[ 0 0 0 0 -1],
[ 1 0 0 0 0]
[ 0 1 0 0 0]
[ 0 0 1 0 0]
[ 0 0 0 -1 0]
[ 0 0 0 0 -1],
[ 1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 -1 0]
[ 0 0 0 0 -1]
]
The surface kernel generators from the first block of Magma code above were [ G.2 * G.5 * G.6, G.1, G.1 * G.2 * G.4 * G.6]. Therefore, we obtain the following matrix surface kernel generators:
>GL5K:=GeneralLinearGroup(5,K);
>MG:=[GL5K!MG[i] : i in [1..#MG]];
>rho:=hom< G -> GL5K | MG>;
>A:=rho(G.2 * G.5 * G.6);
>B:=rho(G.1);
>C:=rho(G.1 * G.2 * G.4 * G.6);
> Order(A);
2
> Order(B);
4
> Order(C);
8
> A;
[-1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 0 1]
[ 0 0 0 1 0]
> B;
[ z 0 0 0 0]
[ 0 0 0 1 0]
[ 0 0 0 0 -1]
[ 0 1 0 0 0]
[ 0 0 1 0 0]
> C;
[ z 0 0 0 0]
[ 0 0 0 0 1]
[ 0 0 0 1 0]
[ 0 -1 0 0 0]
[ 0 0 1 0 0]
We use these matrix surface kernel generators to obtain candidate polynomials.
> Mmats,Gmats,Q:=RunGivenSKMatrixGenerators(64,5,[A,B,C]);
Set seed to 0.
Character Table of Group G
--------------------------
-----------------------------------------------
Class | 1 2 3 4 5 6 7 8 9 10 11 12 13
Size | 1 1 2 4 4 4 4 4 8 8 8 8 8
Order | 1 2 2 2 2 2 2 4 4 4 4 8 8
-----------------------------------------------
p = 2 1 1 1 1 1 1 1 2 5 3 5 8 8
-----------------------------------------------
X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 + 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1
X.3 + 1 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 -1
X.5 0 1 1 1 1 -1 -1 -1 -1 -I 1 I I -I
X.6 0 1 1 1 1 -1 1 1 -1 I -1 -I I -I
X.7 0 1 1 1 1 -1 1 1 -1 -I -1 I -I I
X.8 0 1 1 1 1 -1 -1 -1 -1 I 1 -I -I I
X.9 + 2 2 2 -2 -2 0 0 2 0 0 0 0 0
X.10 + 2 2 2 -2 2 0 0 -2 0 0 0 0 0
X.11 + 4 -4 0 0 0 2 -2 0 0 0 0 0 0
X.12 + 4 -4 0 0 0 -2 2 0 0 0 0 0 0
X.13 + 4 4 -4 0 0 0 0 0 0 0 0 0 0
Explanation of Character Value Symbols
--------------------------------------
I = RootOfUnity(4)
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep [1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
[2] Order 2 Length 1
Rep [ 1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 -1 0]
[ 0 0 0 0 -1]
[3] Order 2 Length 2
Rep [ 1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 1 0]
[ 0 0 0 0 1]
[4] Order 2 Length 4
Rep [ 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]
[5] Order 2 Length 4
Rep [-1 0 0 0 0]
[ 0 1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 1 0]
[ 0 0 0 0 -1]
[6] Order 2 Length 4
Rep [-1 0 0 0 0]
[ 0 1 0 0 0]
[ 0 0 1 0 0]
[ 0 0 0 0 1]
[ 0 0 0 1 0]
[7] Order 2 Length 4
Rep [-1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 0 1]
[ 0 0 0 1 0]
[8] Order 4 Length 4
Rep [-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]
[9] Order 4 Length 8
Rep [z^16 0 0 0 0]
[ 0 0 0 1 0]
[ 0 0 0 0 -1]
[ 0 1 0 0 0]
[ 0 0 1 0 0]
[10] Order 4 Length 8
Rep [ 1 0 0 0 0]
[ 0 0 -1 0 0]
[ 0 1 0 0 0]
[ 0 0 0 -1 0]
[ 0 0 0 0 1]
[11] Order 4 Length 8
Rep [-z^16 0 0 0 0]
[ 0 0 0 1 0]
[ 0 0 0 0 1]
[ 0 1 0 0 0]
[ 0 0 -1 0 0]
[12] Order 8 Length 8
Rep [-z^16 0 0 0 0]
[ 0 0 0 -1 0]
[ 0 0 0 0 1]
[ 0 0 1 0 0]
[ 0 1 0 0 0]
[13] Order 8 Length 8
Rep [z^16 0 0 0 0]
[ 0 0 0 0 -1]
[ 0 0 0 -1 0]
[ 0 1 0 0 0]
[ 0 0 -1 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, 0, 0 ]
S_1 =[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0 ]
I_2 =[ 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
S_2 =[ 1, 0, 0, 2, 0, 1, 1, 0, 0, 1, 0, 1, 1 ]
H^0(C,2K)=[ 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1 ]
I_3 =[ 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0 ]
S_3 =[ 0, 1, 1, 0, 2, 0, 0, 1, 1, 0, 5, 1, 1 ]
H^0(C,3K)=[ 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 1, 1 ]
I2timesS1=[ 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0 ]
Is clearly not generated by quadrics? false
Matrix Surface Kernel Generators:
Field K Cyclotomic Field of order 64 and degree 32
[
[-1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 0 1]
[ 0 0 0 1 0],
[z^16 0 0 0 0]
[ 0 0 0 1 0]
[ 0 0 0 0 -1]
[ 0 1 0 0 0]
[ 0 0 1 0 0],
[z^16 0 0 0 0]
[ 0 0 0 0 1]
[ 0 0 0 1 0]
[ 0 -1 0 0 0]
[ 0 0 1 0 0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 1
Multiplicity 1
[
x_1^2 + x_2^2 + x_3^2 + x_4^2
]
I2 contains a 1-dimensional subspace of CharacterRow 4
Dimension 2
Multiplicity 2
[
x_0^2,
x_1^2 + x_2^2 - x_3^2 - x_4^2
]
I2 contains a 1-dimensional subspace of CharacterRow 6
Dimension 1
Multiplicity 1
[
x_1*x_2 - z^16*x_3*x_4
]
We see that the first and third isotypical subspaces of \(S_2\), corresponding to the characters \( \chi_{1}\) and \( \chi_{6}\) respectively, each contain a single quadric. Thus, these yield the polynomials
\[ \begin{array}{l} x_1^2 + x_2^2 + x_3^2 + x_4^2 \\ x_1x_2 - ix_3x_4 \end{array} \]The second isotypical subspace, which corresponds to the character \(\chi_4\) yields a polynomial of the form
\[ \begin{array}{l} c_1x_0^2+ c_2(x_1^2 + x_2^2 - x_3^2 - x_4^2) \end{array} \] Assume that \(c_1\) and \(c_2\) are nonzero. After scaling \(x_0\) and dividing, we may assume that \(c_1=c_2=1\).Magma V2.20-3 Fri Mar 18 2016 10:13:21 on Fordham-David-Swinarski [Seed =
3580857993]
Type ? for help. Type -D to quit.
> K<i>:=CyclotomicField(4);
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> X:=Scheme(P4,[x_1^2 + x_2^2 + x_3^2 + x_4^2, x_0^2+x_1^2 + x_2^2 - x_3^2 - x\
_4^2,x_1*x_2 - i*x_3*x_4]);
> IsSingular(X);
false
> Dimension(X);
1
> HilbertPolynomial(Ideal(X));
8*$.1 - 4
2
> A:=Matrix([
> [-1,0,0,0,0],
> [0,-1,0,0,0],
> [0,0,-1,0,0],
> [0,0,0,0,1],
> [0,0,0,1,0]
> ]);
> B:=Matrix([
> [i,0,0,0,0],
> [0,0,0,1,0],
> [0,0,0,0,-1],
> [0,1,0,0,0],
> [0,0,1,0,0]
> ]);
> Order(A);
2
> Order(B);
4
> Order( (A*B)^(-1));
8
> GL5K:=GeneralLinearGroup(5,K);
> IdentifyGroup(sub<GL5K | A,B>);
<64, 32>
> Automorphism(X, A);
Mapping from: Sch: X to Sch: X
with equations :
-x_0
-x_1
-x_2
x_4
x_3
and inverse
-x_0
-x_1
-x_2
x_4
x_3
> Automorphism(X, B);
Mapping from: Sch: X to Sch: X
with equations :
i*x_0
x_3
x_4
x_1
-x_2
and inverse
-i*x_0
x_3
-x_4
x_1
x_2