curve #85

y² = x³ − 12300937281145149633363x + 178186913040613669561994205239138

a-invariants: [0, 0, 0, -12300937281145149633363, 178186913040613669561994205239138]
rank (lower bound): ≥ 17
conductor (N): 1482227606068422543624563099019961426673156424163925520
naive height: 164.2055
Faltings height: 11.7446
discriminant (Δ): 105406467094579591296588247874235886027076144817990449651642497766400
primes of bad reduction: 2, 3, 5, 11, 19, 29, 43, 379, 683, 77184794499853885577, 395346588657568538203
regulator: 1817695003829779.04293109872927351022456096418502780356708712404856098214
submitted by: David Renshaw
last updated: 2026-07-01 22:35:40

Witness: 17 independent points

(12769374094, 4815991895126940)
(125820594673, 24946324639382916)
(-103323838141, 18603782679944990)
(-26083877519, 21938468531054724)
(-11082842866, 17696185154606860)
(13821257509, 3288271100958360)
(10629645547, 6973761305127870)
(105648047179, -7603198829438850)
(-109627526642, 14463170264818836)
(14233545121, 2446343252265924)
(157251351019, -46177454812786110)
(-22414544291, 21039141738696840)
(-22992476591, -21186336664692540)
(-46119352946, -25444106607377700)
(197214452197, 73638526067350200)
(3422924509, -11667126369923640)
(315386379763, -166342028869106046)

Commentary

New curve of rank >= 17 and naive height 164.205, found as the specialization T = 533 (i.e. t = 533/2) of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique definie sur Q de rang >= 22' (Acta Arith. 82 (1997), roots {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve over ~1.9M specializations and certified by Neron-Tate height pairing.

last edited by David Renshaw at 2026-06-11 15:11:12 · history