curve #202

y² + xy = x³ − 7699301552946471111x + 8260370960120371675267448841

a-invariants: [1, 0, 0, -7699301552946471111, 8260370960120371675267448841]
rank (lower bound): ≥ 13
conductor (N): 377567090887779694323895274233922490
naive height: 142.0857
Faltings height: 9.6722
discriminant (Δ): -266808849431496166419387240071765982358706830625053900800
primes of bad reduction: 2, 3, 5, 7, 11, 13, 137, 91773698536932579421526229059
regulator: 6519710445.716869994962627499550559448523343955253137275995743738
submitted by: Seewoo Lee
last updated: 2026-07-01 22:45:38

Witness: 13 independent points

(38654232246, -7580643190350363)
(203345746649964702/22819729, 87698553847271448902404101/109009845433)
(6220241216550/49, 15509840583005224947/343)
(110996977707102/82369, 430829851308717157515/23639903)
(3833518434406794/2122849, 48470044901113166359581/3092990993)
(9919516408158, -31241782325935811763)
(143201575382181750/1229881, -54175148608867446972558159/1363938029)
(1006894711087356/32041, -31829950172644810031097/5735339)
(11127316314114/625, -36692149724825773563/15625)
(1893419200260/169, -2532103404406521687/2197)
(51182275986198/6241, -347474912239071226197/493039)
(370092489225438/72361, -6237092781954126380967/19465109)
(4632706890, -268365492663219)

Commentary

Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[-72, 120, 144, -48, -24, -120] with shift t=1516/7: 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 16:10:34 · history