curve #750

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y² = x³ − 32649958692867379691275027686697597452x + 114549132270767603843768274723786312715464491851263429204

a-invariants

[0, 0, 0, -32649958692867379691275027686697597452, 114549132270767603843768274723786312715464491851263429204]

rank (lower bound)

≥ 24

torsion subgroup

trivial

conductor (N)

3167944256312105589192172798619147853260732573731118451476838945462641112466843594300635282197096040

naive height

271.6843

Faltings height

20.6745

discriminant (Δ)

-3440937438703068968684090868085792603456772255093702604465517098908181497235811840986312588565292106033087468000000

primes of bad reduction

2, 3, 5, 7, 13, 17, 29, 61, 27917, 25348867, 1794770750633232006834898747541241599, 2531748732023294652052588230419412211917443

regulator

236403719584611619017433856.76443039660508056189183059582760711018450513755793301356398

submitted by

Valery Asiryan

submitted at

2026-09-16 19:05:30 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-16 19:35:07 UTC

Witness: 24 independent points log in to add more points →

(-5619538686295238007, -10980264915434569640811260895)
(-4958460290803274922, -12431103032418673014444523050)
(-5700732666740749449807/1156, -490328509293780095741910034512375/39304)
(-4485259444357636422, -13067525644905358550484486450)
(31554822478559844033/5476, -4333453232129446781558573540523585/405224)
(2507874038851702218, -6959903175739408001624660430)
(2887516644179642490, -6659371731474019004188275918)
(2925693784339041378, -6638400838235798927639514450)
(3294957329804668578, -6537692187366285229865946450)
(4528486970961636478, -7717563195882438222881285750)
(5137848816937602060, -9078815770072234629304901922)
(5426525800142709498, -9857443041865961707801020210)
(2610357230433880352938/441, -105052747210941831098750617711670/9261)
(6584298767856110550, -13602245765294747480519631402)
(70018175976393160886536744150/9824972641, -15206240780953158135103984742171350938804278/973861113148561)
(667851180087847327602/49, -16093492120009975318996829778270/343)
(60670910042735168958, -470596216016083454868892769910)
(110944670598686266578, -1167081440524606432050260260050)
(4761307941001969949553/16, -328481014931778031345356184727415/64)
(374537772689708720580, -7247585737463636417340742252362)
(769239689243512935628, -21334408102179131433721243371550)
(1247952848653737486553, -44085190869867927325562933839975)
(1451370384383227100553, -55292220054306585930389505306975)
(2704275344818780993098, -140629163923592540217476272002510)

Commentary

Specialization T=-83/111 of the family identified by wgxli in the commentary to curve #302. The points were obtained by specializing reconstructed generic sections and running minimized quartic searches. The 24 listed points are certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 19:05:30 UTC · history