curve #3611

y² = x³ + x² − 172376349302545957827814820x + 873457224134517779658354151991968209600

a-invariants: [0, 1, 0, -172376349302545957827814820, 873457224134517779658354151991968209600]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 11338339931713010411611348392408157126038267924244707240
discriminant (Δ): -1781624571339422345390780819676060745406460628585776253945779892186304220000000
Faltings height: 13.8823
naive height: 192.8542
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 37, 47, 1741603, 2235167387686907731, 63093275975701275781
regulator: 748024566770250124070.54905190318539547517324036539829274548957065398973306
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:18:59 UTC
last updated: 2026-09-29 19:18:59 UTC

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

(8985215258335, 7073317937201238450)
(862709029285510, 25336493662115431545750)
(-259050565264982942/17161, 13449003475144564004511834/2248091)
(-115787254097640215/30976, 208708762100710371709502475/5451776)
(-101826874049507585/16129, 84707156827169875967834250/2048383)
(-26277085256723510/1849, 1692077074025939564213250/79507)
(122690600837590/9, 877806540992762116750/27)
(286710121143490/49, 2769004436786593656750/343)
(33346168465146835/3721, 1579091328333386815720500/226981)
(87610291621368595/22201, 52789918327365046234155840/3307949)
(100544088653745190/12769, 3010756599480993844425750/1442897)
(207830835291651550/10609, 77380736222450965570193070/1092727)
(245875510205750590/2209, 121107165875964590609537250/103823)
(362291090397319435/58081, 89549193950693773022727000/13997521)
(6082226991263443570/769129, 1484343333549000514297440810/674526133)
(9227201297095422190/1495729, 12248852766073563539782332450/1829276567)
(18018434717759791540/877969, 63615266134756423388753797800/822656953)
(51280718647330055290/7038409, 38758562092114711445930695650/18672899077)

Commentary

Specialization at T = 9 of the rank-17 fibration of X1092, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #1842 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:18:59 UTC · history