# Sage Days 16 Project Idea Page

## Implement evaluation of elliptic curve isomorphism at a point

PEOPLE: William Stein

sage: E = EllipticCurve('37a'); F = E.quadratic_twist(-17) sage: K.<a> = QuadraticField(-17); E = E.change_ring(K); F = F.change_ring(K) sage: phi = E.isomorphism_to(F); phi(E.gens()[0]) Traceback (click to the left for traceback) ... NotImplementedError: not implemented.

## Create a Cython class for points on elliptic curves and optimize basic arithmetic

PEOPLE: William Stein

- Right now basic arithmetic on elliptic curves is way too slow. It could be sped up by moving the point class to Cython, and possibly by using better formulas for arithmetic, e.g., using projective coordinates.

## Create elliptic curve classes for elliptic curve models in the Explicit-Formulas Database

PEOPLE: David Kohel

- In order to optimize and compare arithmetic, we should first implement alternative models and verify relative performance. The isomorphisms between different models should also be implemented, and classes for isogenies of these models developed, making use first of the new isogenies code, and eventually putting in place special optimized code for specific models.

See the EFD: http://www.hyperelliptic.org/EFD/

## Rewrite abelian groups (hard)

PEOPLE: William Stein

It would be possible to use trac 5882 to rewrite abelian groups natively in Sage (not using GAP), in a way that is much more flexible than the current implementation. This could be useful for many number theory applications. This project is "hard", since many have tried and always failed.

## Optimize/better document/generally improve graph theory library in Sage

PEOPLE: Robert Miller

Ticket #6085 contains a lot of work so that a graph created by Graph(implementation='c_graph') is just as functional as a Sage graph. I will be sporadically working on improving documentation and optimizing graphs all week, and anyone interested is welcome to join.

## Cliquer SPKG for Sage

PEOPLE: Robert Miller, Nathann Cohen (remotely)

## Take a look at the possibility of making GAP a dynamically loadable library

PEOPLE: Robert Miller, Martin Albrecht (hopefully)

## Python implementation of Ford-Fulkerson algorithm

PEOPLE: Robert Miller

I plan on at least copying the Python implementation on wikipedia, since now we have nothing at all for max flow problems. Hopefully then someone who really cares about it will try to use it, realize it is slow, start improving it, etc. etc. etc.

## Frobenius number and genus of numerical semigroups using toric Grobner bases

PEOPLE: Bjarke Hammersholt Roune

- I plan to code Frobenius number (largest gap) and genus (number of gaps) functions for numerical semigrups using two related algorithms based on toric ideals. These algorithms can handle random numerical semigroups generated by numbers with thousands of digits, as long as there are not too many minimal generators. I'm happy to explain either algorithm if you want to help or are just curious. The steps needed are these:
- Find the best way to compute toric Grobner bases in Sage (4ti2?)
- Improve the integration of the library Frobby for monomial ideal computations.
- Code the algorithms (should be easy at this point)

## PolyBoRi and Singular

PEOPLE: Martin Albrecht, Burcin Erocal

Both PolyBoRi's (Gröbner bases in GF(2)[x_i]/<x_i^2+x_i>) and Singular's interface need some (engineering) work & documentation.

## Elliptic curves isogenies -- greatly improve usability

PEOPLE: William Stein

Dan Shumow wrote new code for computing isogenies between elliptic curves, but it is still very rough around the edges. Fix it up. See e.g., http://trac.sagemath.org/sage_trac/ticket/6384

## Multimodular reconstruction implementation

- CRT in rings/arith.py is unsanely slow
MultimodularBasis class is broken

Fix / Unify code / optimize the hell out of it

Add the early termination technique for the reconstruction of multiple values Apply it to Burcin's computation of nullspace over Z[x]

PEOPLE: Burcin Erocal, Clément Pernet

## Minpoly over small finite field

#6296 Implement a certified computation of minpoly over a small finite field.

PEOPLE: Clément Pernet

## Refactor symbolic functions

Merge the `PrimitiveFunction` and `SFunction` classes in `sage.symbolic.function` and make it easier to define new special functions with custom evaluation, printing, etc. methods. This also involves adding capability to represent symbolic sums and integrals in Sage as a first step to adding native Sage code to deal with these.

PEOPLE: Burcin Erocal

## M4RI: improvement of PLUQ factorization

Change/improve the slow columnswap in PLUQ factorization

PEOPLE: Martin Albrecht, Clément Pernet

## LinBox wrappers

Rewrite LinBox wrappers using C++ wrappers of Cython directly

PEOPLE: Burcin Erocal, Clément Pernet

## Ray class groups and Groessencharacters

Implement computations of the ray class group modulo an ideal of a number field, and the corresponding character group.

PEOPLE: David Loeffler