curve #3173

y² + xy + y = x³ + x² − 7573482228526228645786413x + 7025216436568995568432843580877282531

a-invariants: [1, 1, 1, -7573482228526228645786413, 7025216436568995568432843580877282531]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 10598095089074167643269066006409167869829050
discriminant (Δ): 6480625236772527124768545202714046000730979877954336231865644323194216000000
Faltings height: 13.2742
naive height: 183.4737
primes of bad reduction: 2, 3, 5, 7, 13, 31, 61, 73, 163, 173, 193, 211, 229, 997, 1163, 1723, 2269, 6959, 21871
regulator: 338468138323.7336819266498090206996937997048462460345251976100981
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:48 UTC
last updated: 2026-09-26 20:40:48 UTC

Witness: 13 independent points log in to add more points →

(3394161830119, -4519013963505250012)
(2281504290385, 1273620243340930682)
(2343530248735, 1465438682862181532)
(2663724915535, 2398293466180219532)
(3799553023235, 5753432586059784732)
(2541371007235, 2047387840747784732)
(2001932334308355/2401, -133632559850568431265472/117649)
(-49513533967159357958712465/42963790574329, -1062047944710391852496382461213657476764/281613769910371086733)
(5153132332915470135/4695889, -1924754850355559280843986004/10175991463)
(-111530423318561/36, 187442886950048484817/216)
(3487691321205628960335/689640121, -179348608564771933524881036740508/18110639217581)
(96150190465003111519/87422500, -131910821415416478753434120303/817400375000)
(-2338857394062505/6724, -1709904909677465286160929/551368)

Commentary

2-isogenous to curve#2164 via the rational 2-torsion point x=1566508746135. Independent witness points are the images of that curve's witnesses under the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:48 UTC · history