curve #3685

y² + xy + y = x³ − x² − 1243763883354263091668421608561x + 532688172077316525433436028544237463753530401

a-invariants: [1, -1, 1, -1243763883354263091668421608561, 532688172077316525433436028544237463753530401]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 95872894991771433812104612981479282
discriminant (Δ): 555593375466277202100027124421552931318124517202290536775490822304701580710760091733786624
Faltings height: 16.0958
naive height: 219.5007
primes of bad reduction: 2, 3, 7, 11, 13, 17, 19, 29, 37, 41, 71, 101, 113, 251, 467, 2423, 7457, 218191
regulator: 44363486949.69312922174808244502999394905507314224938794518
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:32:50 UTC
last updated: 2026-09-29 19:32:50 UTC

Witness: 10 independent points log in to add more points →

(-721122679420513, 32474571970896280104564)
(676534503526967, 942396679375966862004)
(689781376045847, 1720420381425131473044)
(-1116499359109385389561/1203409, 39339637971987601568404710127860/1320139673)
(-3954145313922681/25, 3366777851546307588571812/125)
(730610578829108231/625, 406301702175705239745659796/15625)
(886791538033151759/1849, 542115465313604186193114540/79507)
(145238038976986662779/237169, 95243017038484714043532135660/115501303)
(1173820448894283788903/1745041, 1489481265745030271069768783988/2305199161)
(2149845742600104296091/3214849, 205070003524033720930962100660/5764224257)

Commentary

Specialization at t = 610/833 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), 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 19:32:50 UTC · history