curve #198

y² + xy = x³ + x² − 1519707018181529074677x + 22869247242133051599466107911241

a-invariants: [1, 1, 0, -1519707018181529074677, 22869247242133051599466107911241]
rank (lower bound): ≥ 14
conductor (N): 219931200577530279341714779925105066004870
naive height: 157.9378
Faltings height: 10.9754
discriminant (Δ): -1311295550412181298956751711791068502351802457869713490554687500
primes of bad reduction: 2, 3, 5, 7, 13, 17, 31, 67, 163, 2671, 19213, 272760893775113648578459
regulator: 57498099076.9160905642634205944578651803129223187865761115602274920
submitted by: Seewoo Lee
last updated: 2026-07-01 22:44:38

Witness: 14 independent points

(29428257775715296/235225, -4827910858317241171052311/114084125)
(388726833316031142/13256881, 90552776990454414052053489/48268303721)
(7911228532252973/258064, 295806306648889730642983/131096512)
(121528141873272/3721, 644932695710007776079/226981)
(12410076961752/361, 23007869174866244481/6859)
(35203694052, 3605240641688769)
(47187126232, 7498420893276259)
(84857410752, 22471059109025379)
(109029826357, 33959748881502634)
(1070008775432, 1106105206155389859)
(1212557108788/9, -1284613342747699957/27)
(37891831182, -4437305361723291)
(33961043607, -3229175523309741)
(28478113232, -1639127433627741)

Commentary

Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-756, 60, 210, 486, 654, -654] with shift t=1788/5: 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 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 16:10:14 · history