curve #3696

y² + xy = x³ − 1018145777293182878109553955338x + 392056528965806426836847991060069035062432292

a-invariants: [1, 0, 0, -1018145777293182878109553955338, 392056528965806426836847991060069035062432292]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 749928839103528874724228030523150
discriminant (Δ): 1145596260455490769045145144217755233360347118938286663380898993069794667766281666560000000
Faltings height: 16.0999
naive height: 218.9002
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 41, 47, 53, 71, 151, 179, 449, 4157, 10357, 44893
regulator: 459310112709.9262660996419465318418409049950605092096836176
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:45:05 UTC
last updated: 2026-09-30 01:45:05 UTC

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

(371861081364212, 8054125242692485870694)
(5517824754396212, 403450057460499722077094)
(-12948353606636846483404/33535681, 5238534675334915492938634362616730/194205128671)
(71090881864033179572/16129, 584895445527350229162003776090/2048383)
(96632602388112567028/208849, 427092503018055597087455189302/95443993)
(2330022222609685074932/5202961, 60415883216539884498874733614454/11867954041)
(2698276019568020256692/961, 140161756796921771756261787243674/29791)
(6008401135270864058068/18809569, 813406483556982587272022032240742/81577100753)
(9261832633174940984276/14280841, 114852624593541387949737490362370/53967298139)
(432453519596754366175028/141300769, 270538004428929692879016709805530682/1679642241103)

Commentary

Specialization at t = -7064/8413 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 2026-09-30 01:45:05 UTC · history