curve #87

y² + xy + y = x³ + x² − 128067973760200094197595276x + 551283524310015496401062930694209419949

a-invariants: [1, 1, 1, -128067973760200094197595276, 551283524310015496401062930694209419949]
rank (lower bound): ≥ 18
conductor (N): 264272322302981023294226801215719105908812968488670879510837270
naive height: 191.9574
Faltings height: 13.8687
discriminant (Δ): 3141026088650294736579116234013850169228851134468835623113068971509123596492800
primes of bad reduction: 2, 3, 5, 7, 13, 31, 70616456303529097632401, 44220273093899778186557524292644529
regulator: 94146143937784834.222309731780565912552858055405979898830305053693311440478
submitted by: David Renshaw
last updated: 2026-07-01 22:32:41

Witness: 18 independent points

(-12733690254771, -10832050092834520535)
(7116342595149, 544176015111233065)
(5775628631469, 2067176413862584585)
(21771810757029, 89906083538801270665)
(5328143734701, -4492244980411323575)
(7135632829749, -874262666995103735)
(5162168706573, -5266541863119888791)
(8183352551707, 7160640722003337311)
(10353533303901, 18307950595024354825)
(346395899469, -22515836098862263415)
(7803676630173, 5206323527832720217)
(-2691447716443, -29605322946651800879)
(13890747713721, -38112777466974057287)
(22058269995549, -91973691550298533415)
(65965817063413/9, -68106097920473546893/27)
(-10499801042931, 27173753106374947945)
(-12769843462147, -10214059849860638471)
(22805787632961, -97426767137009707655)

Commentary

New curve of rank >= 18 and naive height 191.96, found as the specialization T = 679/26 (t = 679/52) 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), sextuple {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve over ~1.9M specializations; 18 independent points certified by positive-definite Neron-Tate height pairing.

last edited by David Renshaw at 2026-06-11 15:35:46 · history