curve #2003

y² + xy = x³ − x² + 861720912597366549278406x + 6486831363114452406834443307946544100

a-invariants: [1, -1, 0, 861720912597366549278406, 6486831363114452406834443307946544100]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 74982942042145889714439140878353729492075810 ☆ record for ℤ/2ℤ torsion, rank ≥ 15
discriminant (Δ): -18219072298098297211264257907019266845043091812417972706067593972414193750000 ☆ record for ℤ/2ℤ torsion, rank ≥ 15
Faltings height: 13.3122 ☆ record for ℤ/2ℤ torsion, rank ≥ 15
naive height: 183.0488 ☆ record for ℤ/2ℤ torsion, rank ≥ 15
primes of bad reduction: 2, 3, 5, 7, 13, 41, 47, 61, 137, 167, 293, 1223, 1367, 1439, 2243, 2557, 3253, 90289
regulator: 83512415414065071.060441295906557782919839627607993232166565220698597
submitted by: Steps Unbounded
submitted at: 2026-09-26 04:58:02 UTC
last updated: 2026-09-26 04:58:02 UTC

(-1711287569684, 25781128664104842)
(-42478999904756/25, -42739822363394984382/125)
(-208740995762312931/124609, -25741517272681115033102991/43986977)
(-11587928757720675/6889, -301464243620146653598965/571787)
(-123764107986376899/72361, -1898112002502065886416622/19465109)
(-534524430676497183764/315453121, -2251602451055429948029053663798/5602762882081)
(-2467571223618516884/1442401, -133714055354737556007488758/1732323601)
(-314037229604957300/190969, -65865201895990985960968190/83453453)
(-1613739778878479411/946729, -235758706631616057831934411/921167317)
(-1158235825865075675/683929, -233629860696592498138595565/565609283)
(-3001816612380844064/1755625, -282374885626379906981799666/2326203125)
(-4363301400316471682144/2552371441, -17207970999202595477790923127738/128948357570761)
(-880136234129805826976/516062089, -2785846424488849608686222867154/11723382475813)
(-4223590951466520765824/2523153361, -75398977946410174255738188053978/126740516476391)
(-1945746221984182736743011/1154169908329, -611046886794435965358139526651455986/1239951278425736267)

2-isogenous partner of curve#1976 (Elkies (2009)).

last edited by Steps Unbounded at 2026-09-26 21:37:01 UTC · history