curve #190

y² + xy = x³ − 462840480498050459959650246x + 3815301189936260852812257361005424248676

a-invariants: [1, 0, 0, -462840480498050459959650246, 3815301189936260852812257361005424248676]
rank (lower bound): ≥ 16
conductor (N): 105796999633828233559899461972111361932925512955353458023082540370
naive height: 195.8119
Faltings height: 14.1493
discriminant (Δ): 57200821486488614373193385715411199336388466289969673621065223447387681934540800
primes of bad reduction: 2, 5, 13, 17, 31, 59, 89, 157, 10687, 4882196499566433409319, 2761616040077202452279063069
regulator: 214216026004757241.3332241685125651774174884966523042487861989266602557
submitted by: Seewoo Lee
last updated: 2026-07-01 22:40:39

Witness: 16 independent points

(469312363122379396/2401, -319638192435662415699200342/117649)
(75846233912356206459580/680844649, -20524335079905029521908270923524558/17765279426357)
(1407409071959436/25, -49399211055052979742974/125)
(28213970481676, -114960214643784858518)
(8381669212173592671196/573842025, -179793433897362095540192541040034/13746385708875)
(3005595537603714604/227529, 273724828703003818689318386/108531333)
(6603562616910707537996/497334601, 36247012241943409079347316562962/11091058936901)
(8775236465006116/625, 143764105112299072042714/15625)
(70619085957476, 568604587009163644042)
(72904017699180, 597960596232290981546)
(1201852798686484/9, 41155073865803806865014/27)
(442092042524249452/1089, 293542342404627768229271914/35937)
(1193902467929398508/22801, 1208195596369186292976575590/3442951)
(470290282919339747660/30791401, -3003906506723684743914116368974/170861484149)
(3239968495326088/81, -161864274616498478591270/729)
(841154609703266, -24387823893121165826928)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=1069/4: 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 16 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 16:09:36 · history