curve #3645

y² + xy + y = x³ − x² − 460855466999674963016476998152x + 114346210304297863156871290426720957935291979

a-invariants: [1, -1, 1, -460855466999674963016476998152, 114346210304297863156871290426720957935291979]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 12334433622321215895724716304610404854359954238330
discriminant (Δ): 615899793300954530501810881212149850519364092250613667385586567435386111486671360000000000
Faltings height: 15.9834
naive height: 216.5222
primes of bad reduction: 2, 3, 5, 7, 11, 37, 43, 59, 67, 97, 137, 151, 157, 449, 631, 11551, 172351, 356533, 66670921
regulator: 1474443188037194718.4180034984213254134674559227298627471641692317271
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:26:06 UTC
last updated: 2026-09-29 19:26:06 UTC

Witness: 15 independent points log in to add more points →

(500377196720737, 3004621962976095962381)
(510232259383039, 3469178130626886039287)
(771611998707149, 14769930934745969270873)
(972614224161487, 24211230454173535299881)
(-16997923422676499677715147/23001665569, 24998400109735179669722872576776781607/3488501605191247)
(-1965470045313711707/11449, 16814923987923146188227278683/1225043)
(-82575272110633243/121, 14024423438722708344713531/1331)
(118962591094178053/169, 25877659521017541478914557/2197)
(415992646427510557/361, 228872960220413725272363779/6859)
(505804140331391767/841, 180417793995221963535548659/24389)
(167051245226477568613/66049, 2087785399375569396967997508433/16974593)
(362706393811116758893/1520289, 7947634909249770703200697117897/1874516337)
(865769947136977192201/5329, 25474165930487603658452034170969/389017)
(2259143602660144643509/7209225, 16210790235559601053595801092253/19356769125)
(3066581192810895907237/6446521, 27205264631705234617670722523099/16367716819)

Commentary

Specialization at T = -419/177 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (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 19:26:06 UTC · history