curve #754

y² + xy + y = x³ − x² − 49583528871388973584910084x + 137350050888045484014272003399865447079

a-invariants: [1, -1, 1, -49583528871388973584910084, 137350050888045484014272003399865447079]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 15251203000259881061249343205416584739119678368456217166
discriminant (Δ): -347941421023127134233199319395414222408456695945653647904217611171809101152256
Faltings height: 13.6622
naive height: 189.1543
primes of bad reduction: 2, 3, 7, 11, 41, 67, 71, 1667, 8677484058375482423, 13132215161784692278819
regulator: 3172353687724361.7199080024659721020772175990326204377515112946053522096870
submitted by: Valery Asiryan
submitted at: 2026-09-18 01:22:54 UTC
last updated: 2026-09-18 01:22:54 UTC

Witness: 18 independent points log in to add more points →

(-8041906180091, -4000987219869556291)
(-7540859247683, -9079909973253166675)
(-6432364117739, -13789424450354780707)
(-5645385141851, -15406088037822495619)
(-4637547827459, -16357167139197747667)
(-2902403497403, -16025346220110127363)
(-2291056120163, -15457135921373099923)
(-1014348599051, -13660210601611071331)
(411889889221, -10816514207464791427)
(1391258042797, -8429679524069823163)
(2225787143671, -6165591523305089845)
(138461435283993913/64, -51521790935383263519176975/512)
(611156309929333, -15107749019070459245155)
(108983282491129/25, -251332603554831050927/125)
(3632416294213, -2273712679048294915)
(62144915370822013/3481, -14438096208143119447004681/205379)
(1633075507096888541/7225, -2085936826191907178827290119/614125)
(87995534283422269/9801, -19787883532142566354031249/970299)

Commentary

Specialization at T = -568/1107 of the rank-11 fibration of X163 described by wgxli for ICARM #718. Independently recovered polynomial sections and rational-point search yield 18 points with an exact independence certificate, proving rank >= 18.

last edited by Valery Asiryan at 2026-09-18 01:22:54 UTC · history