curve #2146

y² + xy + y = x³ − x² − 1018436625192376093620646909828872955787x + 5885345594761984465165172519607401177779796530129606457115

a-invariants: [1, -1, 1, -1018436625192376093620646909828872955787, 5885345594761984465165172519607401177779796530129606457115]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 18331238590268621233314106535710120397922860155863660882325091066952394194
discriminant (Δ): 52642185190662835903453865489574151738698284302595303449862441182888582893882932003889202077851991627389985755557527552
Faltings height: 21.4758
naive height: 281.0709
primes of bad reduction: 2, 3, 43, 59, 71, 83, 151, 157, 163, 233, 331, 503, 617, 1061, 1097, 9109, 43607, 45589, 53831, 290761, 6923687, 322437920267
regulator: 2275832272073902918.66612151415823533442318522418412636015247694194888698
submitted by: Steps Unbounded
submitted at: 2026-09-26 11:54:47 UTC
last updated: 2026-09-26 11:54:47 UTC

Witness: 17 independent points log in to add more points →

(4701737782536591947, 34653453511120231760510654922)
(-7021755092120972053, 112651454525439849773192525130)
(-20849226553494100777, 134372638836755647806121445274)
(265629708239995233859/9, 1058457510596232042484056785102/27)
(75692414394446704907, 602050759353034087042701371850)
(-3897672949780489701997/121, 96598652467233178188612595252542/1331)
(-31118773598535966763, 86273439913626203893149981510)
(-204630933877381575493/49, 34412402987253183723668447893926/343)
(411967444066420099883, 8336922297560136964928309004042)
(21860327056733272260899/4489, 9705344175855551710375757913420078/300763)
(11766656342725593354785155/335241, 22451907647285139900039162067491320846/194104539)
(-23468068566190202773, 129850073934152050084986322890)
(-361419812915535488497129/14161, 208124801578607559760670474494824114/1685159)
(-34005199517294844117, 34575302567553096730668428042)
(79109635355087743891451/49, 22246388494640512817767553307148710/343)
(4807668545457161595, 33168701561131947567501440954)
(42023788095174738737099735/1471369, 17476160475442088811750983132170755678/1784770597)

Commentary

Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=2/5, t=74/4903. Found by a Mestre-Nagao sieve over t=a/b with |a|,b<=30000, screened with PARI ellrank (2-descent): 17 independent points, 2-Selmer upper bound 17.

last edited by Steps Unbounded at 2026-09-26 11:54:48 UTC · history