curve #3630

y² + xy = x³ − x² − 106891136267469572664x + 422858173212173751446470249404

a-invariants: [1, -1, 0, -106891136267469572664, 422858173212173751446470249404]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 2521187658201578085610417610020422
discriminant (Δ): 918187302983119163652784630732987876485707216186970671940372
Faltings height: 10.3400
naive height: 149.9686
primes of bad reduction: 2, 3, 17, 31, 61, 73, 109, 139, 151, 307, 349, 499, 1237, 1637, 3301
regulator: 297746089515754.14561735953734286871690376088536775604632545241
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:22:59 UTC
last updated: 2026-09-29 19:22:59 UTC

Witness: 12 independent points log in to add more points →

(-547226705029308/49729, 5725869786720358335030/11089567)
(8681998641555/1369, 294914630133175899/50653)
(61034424381507/6241, 275760471270464934762/493039)
(442964639418780/57121, 3352100279963233675722/13651919)
(4085055012798975/139129, 246690479062815476918466/51895117)
(5274024958408350/822649, 24490487888777065759014/746142643)
(31421875589512131/8994001, 8183945225246681004875025/26973008999)
(513064421771629197/125193721, 324392402642237420477201592/1400792544269)
(863782513743488985/8404201, 798882495966038615435506188/24363778699)
(1286312806628622219/293471161, 987086543871613227459052098/5027454459091)
(9981033972422058990/1514999929, 3996344551337697480805294326/58968342236467)
(150154328453348563542/9781803409, 1498480740977184729440699777424/967449702560327)

Commentary

Specialization at t = -13, 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 19:22:59 UTC · history