curve #4257

y² + xy = x³ + x² − 281076797462737535986734102x + 1809463511705756359279122892522372552920

a-invariants: [1, 1, 0, -281076797462737535986734102, 1809463511705756359279122892522372552920]
c-invariants: [13491686278211401727363236921, -1563376474113874682064248764652285109999725]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 311560278166741566302026290678974420505803313066
discriminant (Δ): 6762888600989447475499152769013148703872694934096024202673820122626608957947212
Faltings height: 13.9991
naive height: 194.3156
primes of bad reduction: 2, 3, 11, 13, 19, 31, 59, 67, 89, 97, 107, 163, 173, 467, 5209, 5563, 384751, 566173, 2031067
regulator: 1601310897469784.56804067367687244706682997181647939596340914981402
submitted by: The Wren
submitted at: 2026-10-15 20:36:51 UTC
last updated: 2026-10-15 20:36:51 UTC

Witness: 14 independent points log in to add more points →

(-54580233026337/4, 445725131224368435471/8)
(-11228545134402, 59580619171889922690)
(633973247598, 40392117100145095440)
(13633364485931/4, 238803924015930560097/8)
(58056306834859/4, 224485960222295378417/8)
(16623128437749, 41599634417056381392)
(-27266817278144103684/3308761, 359414998087732963406556386940/6018636259)
(-689005390784717/36, 2884388787011578212527/216)
(26136625872186/25, 4868184654943944776226/125)
(251627370729564/25, 29243934411713088852/125)
(35071070909986002/1369, 5413358258681145476633730/50653)
(416171998921244460/57121, 166276641579592681009604100/13651919)
(15685164155695925454/3301489, 144634757112202005160365367632/5998805513)
(23816802130887971693/769129, 101848003554651242357347987191/674526133)

Commentary

Found by specializing the family associated with curve#802 at T = -334/279. The original PARI/GP computation returned rank bounds [14,14]. Fourteen rational points are supplied.

last edited by The Wren at 2026-10-15 20:36:52 UTC · history