curve #196

y² + xy + y = x³ − x² − 585078982552785047x + 179426017231461870343844271

a-invariants: [1, -1, 1, -585078982552785047, 179426017231461870343844271]
rank (lower bound): ≥ 14
conductor (N): 15241162564541931655288262509967985228690
naive height: 134.4268
Faltings height: 9.1303
discriminant (Δ): -1089582079144626137486468331565725618132214569984000000
primes of bad reduction: 2, 3, 5, 13, 17, 5693, 134599091459570158754656517308997
regulator: 418268198207.917346994581019827048869512072621751054302130902941757
submitted by: Seewoo Lee
last updated: 2026-07-01 22:43:38

Witness: 14 independent points

(745371953239691/2088025, 12089821218503314681926/3017196125)
(7029737347501871/18653761, 284338310384923385718546/80565593759)
(840199390735491/1985281, 7718166572582854963326/2797260929)
(15489085411221/32761, 17275921179556260504/5929741)
(23487639989/49, 1035223960079142/343)
(82305778619/169, 6945576670507158/2197)
(384190709219/289, 205579825992367902/4913)
(84104283089/49, 22312839290733942/343)
(229101138094731/96721, 3306517985548651348914/30080231)
(3176482211, -174276552517506)
(-30900100109/81, 13581992367211726/729)
(-2023912553071/6889, 10323306496153093428/571787)
(151595411, 9706343114094)
(1191988196819/5329, 3006911455148593998/389017)

Commentary

Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=267/2: 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 14 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 16:10:04 · history