curve #3687

y² = x³ − 3123539885838916490562652730700x + 2124621830902586944408688768063715547360666000

a-invariants: [0, 0, 0, -3123539885838916490562652730700, 2124621830902586944408688768063715547360666000]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 187455817312613230634302607218785600
discriminant (Δ): 332821590487182885899159211171165207380796777356606522812117012946714310908854364160000000
Faltings height: 16.2126
naive height: 222.2632
primes of bad reduction: 2, 3, 5, 7, 31, 43, 71, 89, 101, 137, 151, 317, 359, 557, 23773, 70121
regulator: 42016251504.34224951999012235546193395594956534102073482765
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:33:10 UTC
last updated: 2026-09-29 19:33:10 UTC

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

(-1950332338601230, 28247050862142209168400)
(-164349407766190, 51317975058955534465200)
(1007861884074245, 544612158496118690775)
(1033634668656845, 598870052946853870275)
(177987925635835520/361, 182046805675652994234687900/6859)
(2158271962342471810/2601, 1353414578148616055548193200/132651)
(37529110310662240370/36481, 1258326583654763301729296400/6967871)
(46068797956252024640/18769, 247243711779369533077512686100/2571353)
(62474820767094228770/121801, 1089932630686338004828193811600/42508549)
(2172885329090136251105/2184484, 4358478099884026353677087122575/3228667352)

Commentary

Specialization at t = -1877/1424 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 19:33:10 UTC · history