Many methods for solving diophantine questions rely on knowing the group of rational points on an abelian variety. In only very few cases this is know to be a solvable problem at all. We do not even have an algorithm that is guaranteed to work for all elliptic curves (one dimensional abelian varieties).

We do have quite a collection of methods for elliptic curves that work quite well in many practical cases. We can also handle Jacobians of hyperelliptic curves in a fair number of practical settings. This motivated us to try Jacobians of non-hyperelliptic curves. The simplest examples occur as genus 3, smooth plane quartic curves.

We have mixed success. There are several theoretical and computational obstacles, although most theoretical obstructions can be replaced with much worse computation ones.

Ingredients we need:

• Gröbner basis and resultant computations (these are not a bottleneck)
• Rings of integers of degree 28 algebras with very large discriminants
• S-unit groups of number fields up to degree 28
• Identification of galois groups and decomposition groups in Sp(6,GF(2))
• Cohomology of several Sp(6,GF(2))-modules over GF(2)
• Computations in p-adic extensions

In other words, we are using a very large part of magma and in many cases (judging from the bugs we ran into), drive it a little beyond its comfort range.

This will be more a progress report than a polished presentation and I will try to emphasise the interaction with magma we experienced.

This is joint work with Victor Flynn, Bjorn Poonen and Michael Stoll.