curve #1924

y² = x³ − 3003148213658533121422467x + 39769767947993501637839342113700187026

a-invariants: [0, 0, 0, -3003148213658533121422467, 39769767947993501637839342113700187026]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 785713583551212949496355574586236811508456240
discriminant (Δ): -681532633395214059646622978148552569586592272696037882869886557847911223680000
Faltings height: 13.6141
naive height: 186.6755
primes of bad reduction: 2, 3, 5, 29, 67, 211, 241, 337, 433, 479, 1117, 1181, 1237, 1637, 1789, 16253, 2034421
regulator: 1938148103456.109795538895289431348778403124930591194661615212888
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-25 19:15:59 UTC
last updated: 2026-09-25 19:15:59 UTC

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

(444736605302228557/361201, -1337102688125678802586983360/217081801)
(39844637988085, 251350881219660263616)
(-2424597658883, 5726934199482483600)
(5281419597255849565/1841449, 18489596476221430608876080688/2498846293)
(-2498701011970, 5627884220649718004)
(4901629494826357, 343171089791001979503840)
(12460730024703037/3721, 1861611474585164225838480/226981)
(5860696183777, -14948946893043763740)
(2764385557057, 7252093904057712540)
(-2732464349983, -5251113878460355100)
(8519948894462, 25152379159784227220)
(3015215238517, -7624141144045869600)
(41012346297070, -262487875416090955344)

Commentary

Rank exactly 13, torsion Z/2Z: 13 independent points; 2-isogeny Selmer bound 13, confirmed by a second 2-isogeny descent. Specialisation of Kihara's rank-8 family (S. Kihara, Proc. Japan Acad. 77A (2001) 11-12) with u = (s^2 - l)/(2s), (p,q,s) = (5,4,250/1).

last edited by Alvaro Lozano-Robledo at 2026-09-25 19:15:59 UTC · history