curve #3635

y² + xy = x³ − 1449906033237470410635x + 17073586021555823773300048810225

a-invariants: [1, 0, 0, -1449906033237470410635, 17073586021555823773300048810225]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 8517544174435024143004460515381044430
discriminant (Δ): 69142899231148177935060350089607447124329289768529170022400000000
Faltings height: 11.1570
naive height: 157.7910
primes of bad reduction: 2, 5, 31, 151, 193, 199, 241, 311, 409, 1031, 1721, 2333, 2437, 2711, 5651
regulator: 834237764991718.7028386439467062690949270563631122265203157966983
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:23:54 UTC
last updated: 2026-09-29 19:23:54 UTC

Witness: 13 independent points log in to add more points →

(642998256463902, 16304770252901039521441)
(-1082604861778690/26569, 12626439696697732579755/4330747)
(-1028345520020370/24649, 8609367015441893855045/3869893)
(-36302761587770/8649, 3864715856739211216405/804357)
(774552365845/16, 497106550180425535/64)
(19379923473630/2809, 404954673922084296605/148877)
(52091107629310/9, 375955046155281021115/27)
(2402155348381470/196249, 93641309059065339504355/86938307)
(71270177161463070/1739761, 11796061363856241089249335/2294744759)
(158193677749124270/5248681, 10432987622587004418638315/12024728171)
(2237332065773628670/69239041, 1146781850352455313960446465/576138060161)
(3783959112265361330/344213809, 10022791775877118878197219505/6386198798377)
(1061577159127196192270/29580216121, 17072488116289920360218086341485/5087471790434669)

Commentary

Specialization at t = -31/84, u = 11/5 of the Elkies-Klagsbrun family y^2 = x^3 + 2A(t,u)x^2 + B(t,u)x (arXiv:2003.00077, Sec. 9, eqs. (3)-(4)), 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 19:23:54 UTC · history