curve #407

y² + xy + y = x³ − 3709951130930525095023568x + 2771978727946269955724382333144546458

a-invariants: [1, 0, 1, -3709951130930525095023568, 2771978727946269955724382333144546458]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 417915012125990192528679544512198522920598025470860459357489652330
naive height: 181.3484
Faltings height: 12.9666
discriminant (Δ): -51411382397974155997137496716460022041538318345893415134584180167571187500
primes of bad reduction: 2, 3, 5, 5711, 7290821, 334563162888224059184705111206367940720505405735115281
regulator: 32782880949544867900.965466547617252390471883374763399879940054415135280760
submitted by: Bhavik Mehta
submitted at: 2026-08-28 20:29:12 UTC
last updated: 2026-08-28 20:29:12 UTC

Witness: 18 independent points

(2130601975627, 2130572130898993337)
(25751724149899231/12769, 2696525099524257749295235/1442897)
(14606543505187554259/654481, 55622719438366410944021466115/529475129)
(30109621143837/121, -1817292188647547371933/1331)
(-92678458986528884/70225, 43122760131099770560951166/18609625)
(-47360087420559161/63001, 35837418091206943031772985/15813251)
(7707330871611488179/7447441, 4112717069346587762867669635/20324066489)
(702781138515934/961, 19984098585996791370065/29791)
(2744907957921668101/2362369, 628639679469781529381272385/3630961153)
(623271490551691/529, 2322646692772128950815/12167)
(4211082445460553484/829921, -8093208125551926636000039125/756058031)
(8118426680105016394/22201, -23131394115574413530320203665/3307949)
(4359600934191271/25, -287835092950230505646471/125)
(31688813019094159/32041, -1519514179290520924897145/5735339)
(30942968920004368964884/1077874561, -5431112771346199973574437432939885/35387699712191)
(22308177456532574119/44903401, -308545565525824024705212447185/300897690101)
(130200220826670817/10816, 46412985402759744291977683/1124864)
(18357118322078602/27889, -3653035558070951451401651/4657463)

Commentary

Rank-18 curve from Mestre's quartic construction, specialized at T = 235/3. The sextuple is [0, 113, 550, 753, 868, 1058], the canonical primitive representative from u = -3, v = -1/2 in the Mestre-Fermigier family, whose generic member has rank 12.

last edited by Bhavik Mehta at 2026-08-28 20:29:13 UTC · history