curve #3780

y² + xy + y = x³ + x² − 5249460128724x + 4502799750307510356

a-invariants: [1, 1, 1, -5249460128724, 4502799750307510356]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 874968923829 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 5
discriminant (Δ): 499245970100099981317154927071636905009 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 5
Faltings height: 6.2007 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 5
naive height: 99.4810 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 5
primes of bad reduction: 3, 7, 11, 13, 19, 23, 29, 83, 277
regulator: 68097.2092076381323085909473017801776941860633872
submitted by: Daniel Błażewicz
submitted at: 2026-10-01 19:48:37 UTC
last updated: 2026-10-01 19:48:37 UTC

(2154478, 1785976841)
(-42338306100/54289, 36055273588597524/12649337)
(253373109448/310249, 150618258172956751/172808693)
(1059889255247/68644, 1079621513283307545/17984728)
(275404626046/251001, 31722561133908584/125751501)

(u,v) = (4067,1196) for the C2 x C4 family y^2 = x(x+u^2)(x+v^2). PARI/GP gives exact rank 5.

last edited by Daniel Błażewicz at 2026-10-01 19:48:37 UTC · history