### 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]`