curve #200

y² + xy + y = x³ − x² − 1213916060668229132x + 555373168020983867224329231

a-invariants: [1, -1, 1, -1213916060668229132, 555373168020983867224329231]
rank (lower bound): ≥ 13
conductor (N): 6177584096452092230279206747860870
naive height: 136.6865
Faltings height: 9.3477
discriminant (Δ): -18761498098102556371487217190377845102127324807168000000
primes of bad reduction: 2, 3, 5, 7, 11, 31, 28755686340139143649765892789
regulator: 12746195036.17266378810682497049084448405859392226472703342881533
submitted by: Seewoo Lee
last updated: 2026-07-01 22:45:37

Witness: 13 independent points

(14353961139/25, 862997361513753/125)
(52759158221/25, 10747244403736929/125)
(1986614540666021/638401, 83677707197040577478091/510082399)
(4289402483179541/1324801, 266593073543863450615179/1524845951)
(1225643375129621/323761, 41279856407830476915741/184220009)
(33314611142315909/8334769, 5872502213681351200460547/24062478103)
(4092206741, 253210335472629)
(1271021187149/289, 1392062438251856277/4913)
(59579399432021/11881, 449676106276783198641/1295029)
(1607845076494709/299209, 63218916112170048001407/163667323)
(5586961541, 410077035801429)
(6838692021, 558641128118069)
(341112840993869/5041, 6299253437563069859739/357911)

Commentary

Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=2289/5: 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 13 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 16:10:24 · history