curve #3655

y² + xy = x³ − 2706381903778836656039400049062720x + 53529269933708805299470502391666979470354686054400

a-invariants: [1, 0, 0, -2706381903778836656039400049062720, 53529269933708805299470502391666979470354686054400]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 4657372890037610572108520563138067741151339367609178987970
discriminant (Δ): 30820409949698128285263975281254460997877068798364715878074874853402716869542741888187690553344000000
Faltings height: 18.0870
naive height: 242.5564
primes of bad reduction: 2, 3, 5, 11, 19, 23, 31, 37, 43, 59, 79, 103, 113, 257, 271, 601, 1051, 2797, 4919, 81013, 270071, 599413, 1520587
regulator: 1102572221540489776321.799250889128892802909038557606138733181273381146
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:27:47 UTC
last updated: 2026-09-29 19:27:47 UTC

Witness: 16 independent points log in to add more points →

(-2444046140789992946886720/143017681, 16651828122563729682860030248908178560/1710348447079)
(-2076774779730084522360/657721, 4201553727509615618081412340404480/533411731)
(-25977384715861011210/3721, 1927108165601262954032226304230/226981)
(2135076990478270689/64, 311745418559039860277318979/512)
(271925412154445852301/10000, 207282615316348628231529371499/1000000)
(108225208733951064524520/2301289, 19204781562837876115380502355953920/3491055413)
(176834239890189215404800/5340721, 5774056458550050561959509574393280/12342406231)
(1775644256142638001637955/1811716, 2362836415839724165861226030852895415/2438569736)
(1575854446583075401302528/45738169, 336445051947159598333610044353507264/309327236947)
(1691720375353861939340160/35129329, 1229655154972523797314657691180319040/208211532983)
(18096175034864370351464880/133148521, 72000571228563946467976336663992531120/1536400783819)
(1731776776819240223845947648/29387216329, 50046035415357575609060600250020858121408/5037762333631483)
(1928799843893888043256928220/142815212281, 237972255189638450521361226188291950906320/53971154057900429)
(1976208709812282482164086240/47777653561, 36703551513464631230946551312248850297280/10443287293016941)
(3272723992107066014754499200/86789749201, 57711903370207855967938473354638658889920/25568346904363801)
(12840002751040210974917164800/392450878681, 17659385451964071789582345819721118819520/245854385007620579)

Commentary

Specialization at T = 477/1130 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 19:27:47 UTC · history