curve #709

y² + xy + y = x³ − x² − 560157780870413809452x + 4662409390554898716115514935215

a-invariants: [1, -1, 1, -560157780870413809452, 4662409390554898716115514935215]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 2163467356080881562813504882331472558986861814
discriminant (Δ): 1858084360591292053554937673443983445315471988796250392102633472
Faltings height: 10.8779
naive height: 154.9379
primes of bad reduction: 2, 7, 11, 13, 17, 37, 59, 228619, 523722169, 243203789580480755194967
regulator: 3897015030497.646183794582934629696977314204559870403789035399794596180
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:43 UTC
last updated: 2026-09-10 16:08:43 UTC

Witness: 16 independent points log in to add more points →

(-5291423307, 2734648128544917)
(-798390763, 2260337823596437)
(17185125333, -333603509892523)
(8336418613, 756333288315189)
(59534702057, -13502890007614423)
(-26844552619, -595473199776299)
(86149312357/9, 11376923345243599/27)
(131851846957, 47149027429084165)
(199482525765, -88492948838729851)
(27747658153, -3237785870206611)
(370181171293/9, -193210780147128041/27)
(10137438869, 160135713201685)
(7057881661, 1029783446968861)
(261038931445, 132838033253200629)
(25970431471, -2762434626019253)
(-19694052779, 2838272585421781)

Commentary

Found by a search of rational specializations of Mestre families, using finite-field scores to select fibers and rational-point searches to construct the witness. Point independence checked with the site verifier.

last edited by David Renshaw at 2026-09-10 16:08:43 UTC · history