implementation in Sage
2-descent
- Start with Sage 3.4.1.rc3 or later.
#5854 - ratpoints
two_isogeny.patch - implements descent via a 2-isogeny
- do a second descent in ambiguous cases
- implement Siksek's tests for large primes
- general case, PHS method
- general case, number field method when only computing the Selmer group
compare (a,b,c) search range with mwrank as a benchmark...
Literature
2-descent
- originally described in BSD's *Notes on elliptic curves, I*.
- improved in Cremona's *Classical invariants and 2-descent on elliptic curves*
- These two are mainly what Cremona's book is based on. And then there are improvements:
- Cremona's *Reduction of binary cubic and quartic forms*
gives much tighter bounds on (a,b,c)-range.
- Cremona and Stoll's *Minimal models for 2-coverings of elliptic curves*
adds conditions to rule out the larger (I,J) pair
- Pascale Serf's thesis
adds divisibility conditions on (a,b,c)-range
- Siksek's thesis
- Describes a better local solubility test for large primes (p. 93)
- gives a new descent method - lattice enlargement algorithm
- algorithm for computing the 2-Selmer group only
- better complexity than PHS method
- discusses higher descent based on the 2-descent when global points are not found
- For curve 174a, invariants method takes forever, but
3-descent by 3-isogeny
- Henri Cohen and Fabien Pazuki's *Elementary 3-descent with a 3-isogeny*
- See Cremona and Roberts' *Report: Applications of polynomial lattices...* (fflatice.pdf) for finding rational points.
3-descent without 3-isogeny
- Djabri, Schaefer, Smart - originally described the number field method
- Schaefer and Stoll - proved the method to work, good exposition
- Stoll's magma code - in email...
p-descent by p-isogeny
for p \geq 5
- Michael Stoll's *Explicit isogeny descent on elliptic curves*