curve #4175

y² = x³ − x² − 910351656599515802377x + 10562164596013837840658404817101

a-invariants: [0, -1, 0, -910351656599515802377, 10562164596013837840658404817101]
c-invariants: [43696879516776758514112, -9125710210693774617228201210890624]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 709486739958211139142167777070531368963706408
discriminant (Δ): 90850671391435354539572872164933853516996075293149197946160384
Faltings height: 10.8041
naive height: 156.3947
primes of bad reduction: 2, 3, 13, 17, 23, 29, 157, 239, 6299, 14891, 4383071508659110290557591
regulator: 110375162066226482.8066887377704413062988676880990852525450322351716488
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-11 22:58:28 UTC
last updated: 2026-10-11 22:58:28 UTC

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

(1854534249, 2979977275888574)
(15969942397, 311223679444686)
(18153508899, 136288197140974)
(57113519905, 12036214552234254)
(155945027676, 60506209143915743)
(747260144796, 645444541958262527)
(-15380765406209735/516961, 1250053223000454730897026/371694959)
(-80494585202903/12321, 5509831701741518719646/1367631)
(-10365054855827/324, 15369349393768130657/5832)
(510754889701/25, 87582159035129226/125)
(591640316385/16, 335376781303025449/64)
(1297763958209/64, 339688216845450369/512)
(38377633027603/49, 237574358556098924802/343)
(227205368036595/29929, 10469684877534481936082/5177717)
(407225388375589/22801, 55377084100175911086/3442951)
(5420529962740541/319225, 1410311883784448191014/180362125)

Commentary

Specialization at T = 7 of the K3 elliptic fibration y^2 + (Lx+B)y = x(x+D)(x+E) given in the commentary of database curve #3857; points from PARI ellrank.

last edited by Andrii Prokhorenkov at 2026-10-11 22:58:28 UTC · history