curve #3620

y² = x³ − x² − 3908966416437847720x + 2875119211480731452257216432

a-invariants: [0, -1, 0, -3908966416437847720, 2875119211480731452257216432]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
conductor (N): 188622615993334511234840880
discriminant (Δ): 251614928158591148671993995923859512114002303073856000000
Faltings height: 9.5904
naive height: 140.0430
primes of bad reduction: 2, 3, 5, 11, 13, 29, 31, 37, 61, 79, 103, 3299, 6337, 15923
regulator: 187905380205.8684821677578431282757231270166893874003236012
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:20:57 UTC
last updated: 2026-09-29 19:20:57 UTC

Witness: 10 independent points log in to add more points →

(878558717, 10908335505054)
(3586143324, 187019594796320)
(68851525844, 18058976726161800)
(-64104617145014/380689, 13952609328653665865910/234885113)
(13881180418921/10609, 798152710794543000/1092727)
(16491463944946/2209, 64819609160516174250/103823)
(47833739116153/49729, 25037118280730392848/11089567)
(227028873825121/235225, 136165707416169560844/114084125)
(2974141203828796/1006009, 132170788764257239900800/1009027027)
(39089448856378308436/40798748169, 25302117525815482726837177000/8240816746411803)

Commentary

Specialization at T = 32/9 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:20:57 UTC · history