curve #3634

y² = x³ + x² − 22546842745086570220760x + 143033305149276646344856595782381908

a-invariants: [0, 1, 0, -22546842745086570220760, 143033305149276646344856595782381908]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 11537416688626259646150573805118160
discriminant (Δ): -8837349834392469949566678205035845402057412418912505963397675421908480000
Faltings height: 12.6760
naive height: 175.4199
primes of bad reduction: 2, 3, 5, 11, 17, 19, 59, 73, 101, 103, 107, 137, 179, 881, 1579, 1979, 2459
regulator: 62325001360551.391531546881102116772510966796813428818874386716
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:23:43 UTC
last updated: 2026-09-29 19:23:43 UTC

Witness: 12 independent points log in to add more points →

(104903799646, 376593284277452880)
(-270698040591600463634/672313041, 5137014128691421509481810751120/17432404840089)
(-30943077278197049684/129709321, 542449701074924114594899355370/1477259456869)
(-26667086420967130658/57592921, 101757046409328853507865226480/437072677469)
(-4527259607401088189/18190225, 28317511515842396816420154384/77581309625)
(-49786524092343641/97969, 4675711847224476841663650/30664297)
(-13920530530152074/25921, 66747725285940548067240/4173281)
(2878265104830937/1024, 154694088218828092949091/32768)
(20054677157801854/49, 2840033637034625535870960/343)
(28208602212434254/12769, 4758191317875770685670320/1442897)
(1272178773606290854/17530969, 27638550757058300754571838280/73402167203)
(10362707801212530784324/352425529, 1054885779006612965798891473557270/6616084455917)

Commentary

Specialization at T = 176 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 19:23:43 UTC · history