curve #3613

y² = x³ + x² − 81321973849053480505925300x + 321025250640720460234849440829085820000

a-invariants: [0, 1, 0, -81321973849053480505925300, 321025250640720460234849440829085820000]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 613662111701893885107735999793593321355118146923962938690680
discriminant (Δ): -10101282699946150736973387858889560363365402529339851351117173755399668531040000
Faltings height: 13.8861
naive height: 190.8523
primes of bad reduction: 2, 3, 5, 11, 13, 19, 53, 71, 40444406887404089, 12367039174898142442876476336022831
regulator: 2443968229711946274117014.458837541608162124677744026085659292826773970095271
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:19:24 UTC
last updated: 2026-09-29 19:19:24 UTC

Witness: 19 independent points log in to add more points →

(-2953783184288383727051525/1255477671361, 31434551286944437739255687830962099000/1406738876684244641)
(-64944718668245545865/7284601, 361153887211264916227485145590/19661138099)
(-684089222834494100/267289, 3128046434867764846023471000/138188413)
(1052360178759916/25, 33421211326032085248264/125)
(60642805546304331/4225, 12617301557664678853176804/274625)
(71398598483474860/6889, 13905860789422558506658680/571787)
(3533850457931517100/537289, 3311020192904233428864187800/393832837)
(4986553838529257215/1681, 11135219040503234885690963910/68921)
(21136233367864131400/5387041, 98732793918997256753104718400/12503322161)
(240619181977233026812/877969, 3730469792722051873934502196992/822656953)
(298372103304871917850/1062961, 5151292885191342348158756487750/1095912791)
(6010253816637724059400/15155449, 465830612846972002147753609308000/59000162957)
(6235584874581412437685/627101764, 349581709171997318709987250909095/15703882374088)
(12937495002834107684355/587044441, 1366686212642430760144361748714120/14223499760989)
(13582952902738805399515/337126321, 1546857319466904786133983790374060/6189976379881)
(37618084646681737366987/15543356929, 22796211450241137113498097242633052/1937836938409217)
(39112145221764286539400/5624850001, 4041037268409196829505607708575600/421858125224999)
(44667067298695942984975/9063230401, 5453234616252507242753744496740250/862828597405601)
(854548567430628169733725/224746157476, 870732001379797264164861192694888125/106546309859277224)

Commentary

Specialization at T = -15/4 of the rank-17 fibration of X1008, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #751 (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 19:19:24 UTC · history