curve #164

y² + xy + y = x³ − x² − 4936153897237243770097313x + 4240417853491998596755638983929308017

a-invariants: [1, -1, 1, -4936153897237243770097313, 4240417853491998596755638983929308017]
rank (lower bound): ≥ 17
conductor (N): 771012142533763749869410487682169741745976568727255733990
naive height: 182.1986
Faltings height: 13.0151
discriminant (Δ): -70418701085637647451617054505242098864199452815445343344812504364494028800
primes of bad reduction: 2, 3, 5, 7, 17, 31, 37, 423061, 148356075867152734210357352683148523657123407
regulator: 11352746991815636.1172596771173279037714821445010842152357302799389226429
submitted by: Seewoo Lee
last updated: 2026-07-01 22:35:42

Witness: 17 independent points

(79842022211577577/65536, -3137523758908713317967093/16777216)
(17605289416521495703/1437601, -72730269278505203939381021988/1723683599)
(63242764747672111/42025, -4042271758419273091936556/8615125)
(26065785237334003/17161, -1117454965119437110793952/2248091)
(1659653179695, -787111906710352892)
(1942280703063, -1407192016767310652)
(2117518569559508823/1087849, -1607545365000713659257625876/1134626507)
(846101499149193063/249001, -642052115317737566574203348/124251499)
(80078043136623, -716316208603657433972)
(70814865534903, 595628366373660638308)
(1128262932027343/81, 37442838388387048510652/729)
(194476369296111/25, 2611815033391840449044/125)
(3517375303503884887/2331729, 1696029492038458545985440364/3560550183)
(10677891747607/9, -6277332203626315604/27)
(6392201212718727/5041, -50731235352220994624260/357911)
(187849311498195823/143641, -8018354400294797441730068/54439939)
(93969411133650535/68121, -4213817666448885805296844/17779581)

Commentary

Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[240, -1692, -996, 1260, 1776, -588] with shift t=1383/4: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 17 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.

last edited by Seewoo Lee at 2026-06-25 16:07:30 · history