curve #214

y² + xy + y = x³ − x² − 579129722683962784448x − 1024153228236700825546996771869

a-invariants: [1, -1, 1, -579129722683962784448, -1024153228236700825546996771869]
rank (lower bound): ≥ 14
conductor (N): 80368938829765696602245459305620130547747310
naive height: 155.0378
Faltings height: 10.9858
discriminant (Δ): 11977921723624303283051037363414017941731155172995286983285145600
primes of bad reduction: 2, 3, 5, 7, 11, 13, 73, 241, 353, 171161, 279750562966837790136450937
regulator: 48717653573893598.9939311056194054995916665713640284929986128650103
submitted by: Seewoo Lee
last updated: 2026-07-01 22:44:38

Witness: 14 independent points

(-12938798579711275/561001, 106534889369693588972853/420189749)
(-9165284686125091/664225, 1127734230889137040381743/541343375)
(-8432396697289099/3916441, 3575196644528288928032739/7750636739)
(-1618070764674451/112225, 78217983332927259567087/37595375)
(-21563290651425/961, 24444873195912878927/29791)
(-10810436653197/841, 50551889024771630845/24389)
(-2622368593781/225, 6869114780706266423/3375)
(-90158812763/9, 52438974274566587/27)
(20952892100429/625, 64874709412740057417/15625)
(100146583014325/729, 986499899603264866051/19683)
(165696032027705/2209, 2017400938636708177491/103823)
(594013394643365/17161, 10154350491931317657147/2248091)
(1300448789723773/25281, 41247706265133603067519/4019679)
(6535886916365997/1681, 528382422141653838127321/68921)

Commentary

Rank ≥14. Mestre construction from the rational 6-tuple family a(u,v) = {u, v, -u-v, -u(u+2v)²/((u-v)(2u+v)), v(2u+v)²/((u-v)(u+2v)), (u-v)²(u+v)/((u+2v)(2u+v))} at (u,v)=(-12,-3), giving a=(-12,-3,15,16,-27/2,-5/2) with shift t=-1161/88 (cleared: tuple (-24,-6,30,32,-27,-5), t=-1161/44). This family satisfies the Mestre S=0 degree-drop condition identically — it is a union of two depressed cubics with equal sum of squares. Certified rank 14 via 14 independent points from an integer quartic-point search + Néron–Tate height-pairing matrix.

last edited by Seewoo Lee at 2026-06-25 21:12:06 · history