Genus 2

Here are some functions for genus 2 curves.  You can run them on our server here.

The input is a genus 2 curve given in Weierstrass equation

    y^2=a_6 x^6 + ... + a1*x+a0;

Enter only   
C:=a_6*x^6 + ... + a1*x+a0;


ModuliPoint(C, x);
The moduli point is defined as follows:


J_2(C,x); 
J_4(C,x); 
J_6(C,x);
J_10(C,x)
Igusa invariants as defined in []

 i_1(C,x), i_2(C,x), i_3(C,x), a_1(C,x(, a_2(C,x),
Absolute invariants as defined in [].





AutGroup(C,x)
Computes the automorphism group of the curve. The autput is given as the GapId for the library of small groups.  


CurvDeg_3_EllSub(C,x)

This function checks if the curve C has a (3, 3)-split Jacobian.  



CurvDeg3EllSub_J2(C,x)

L_3_d(C,x)


Ell_sub, 

J_48, 

LocusCurvesAut_V4, 
LocusCurvesAut_D4, 
LocusCurvesAut_D6, 

LocusCurvesAut_D4_J2, 

Rational_model(C,x)

Mestre(C,x)

An example run in Maple: (There are still some functions which are being implemented). Hopefully will be completed in May 30. 

> with(genus2);
[AutGroup, CurvDeg3EllSub_J2, Ell_sub, Igusa, Info, J_10, J_2, J_4, J_48, J_6, L_2, L_3, L_3_d, LocusCurvesAut_D4, LocusCurvesAut_D4_J2, LocusCurvesAut_D6, Mestre, ModuliPoint, OrderAutGroup, Rational_model, a_1, a_2, autgroup, i_1, i_2, i_3, u_v]

>C:=x^6+x^4-x^2+1;
                         6    4    2    
                        x  + x  - x  + 1
 
>Info(C,x);
                "Initial equation of the curve"
                             6    4    2    
                    "y^2=   x  + x  - x  + 1
                     "Igusa invariants are"
                 [-224, 2128, -140096, -123904]
                    "Clebcsh invariants are"
                       [C2, C4, C6, C10]
              "The moduli point for this curve is"
                    [171  -23787    29403  ]
                    [---, ------, ---------]
                    [28    2744   275365888]
           "The Automorphism group is isomorphic to"
                                             
                 "The automorphism group is V4": [4,2]
                      "Sh-invariants are"
                            [-1, 0]
             "The minimal field of definition is:"
                              "F"
                   "The  field of moduli is:"
                              "M"
                "The  degree of obstraction is:"
                            "[F:M]="
  "Rational model is over its minimal field of definition is:"
          6      5       4       3        2            
         x  + 4 x  + 60 x  - 32 x  + 240 x  + 64 x + 64
    "A minimal rational model is over its minimal field of definition is:"


                      "with moduli point"
                    [171  -23787    29403  ]
                    [---, ------, ---------]
                    [28    2744   275365888]











References:

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