curve #3651

y² + xy + y = x³ − x² − 44700175175144584563197015384837x + 112606878837296599453184780505115482103256620285

a-invariants: [1, -1, 1, -44700175175144584563197015384837, 112606878837296599453184780505115482103256620285]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 373194485014698697229027713931389442079902239969278
discriminant (Δ): 238309517520333455096908703517660122164915444286578516241861572754946754199381531956425079652352
Faltings height: 17.0857
naive height: 230.2462
primes of bad reduction: 2, 3, 7, 11, 17, 19, 31, 37, 59, 109, 167, 181, 283, 983, 2221, 3163, 6563, 7867, 43117, 859378921
regulator: 30866841244057419.312801466735732569602608922663008068324935307995775
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:27:07 UTC
last updated: 2026-09-29 19:27:07 UTC

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

(-6674497450763009, 337070405992539082633472)
(3370317424408375, 15384030974988478275560)
(4712529176184991, 81311270159101668469472)
(5547697914584959, 188055955665835775594240)
(195317847999145459, 86270356969368338446219340)
(-509986188707079721/625, 6021971894336665627147381408/15625)
(-50640995165925689/25, 55176095225381211889288576/125)
(-15645192302294681/9, 11614976509186423313374144/27)
(517116188387614039/169, 147039860879691871186214144/2197)
(963524197526109079/289, 123884510840525079307022056/4913)
(21086036830735608823/3721, 46113459407310498803541749504/226981)
(34596284014333506535/10201, 4197544081312655329120248640/1030301)
(48266542259758564903/10201, 86408116035293547872379104512/1030301)
(54954793261166911517839/31798321, 36092964203786370015008508171573760/179310732119)
(2922556219235105732704285/44209201, 4971612774155090849503519241497919726/293946977449)

Commentary

Specialization at T = -1321/42 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 19:27:07 UTC · history