Rational points on elliptic curves over the rationals
technical explanation
The algorithm computes (exactly) three quantities:
I the rank C of the 2-Selmer group.
I the rank T of the 2-torsion subgroup.
I the rank s of G[2]/2G[4], using the Cassels pairing.
The quantities that we are interested in are:
I the rank R of E(Q)
I the 2-rank of Ø(S) (conjecturally even).
The following formula holds: C = T + R + S.
Here C = 3, T = 0, R = 1, so S = 2. Since s = 0, Ø[2]/2Ø[4]
is trivial so the 4-rank of Øis at least 2, so |Ø| ≥ 16, and we
can conclude under BSD that Ø(E) ∼
= (Z/4Z)2.