curve #1931

y² = x³ + x² − 8522687623987657861532175040x + 782442899642450813837213157949458793913204

a-invariants: [0, 1, 0, -8522687623987657861532175040, 782442899642450813837213157949458793913204]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 587067164462035469998813287245201298336901585011216
discriminant (Δ): -224858133494881927058583899298762867445188269808784134160154648680743417259552643739648
Faltings height: 15.2557
naive height: 206.4496
primes of bad reduction: 2, 3, 11, 13, 17, 19, 29, 31, 43, 53, 59, 71, 73, 151, 421, 443, 14461, 20161, 1498361, 65188429
regulator: 54010090286550.60886559330915630002751629084019540485601072666877
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-25 19:17:22 UTC
last updated: 2026-09-25 19:17:22 UTC

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

(41082966036535566968992535/580066547641, 322867751897191131648100802104673230278/441790864080886061)
(4017446747162509584411412484684/18873209173280401, -7609542045703596937178619640925495484608557602/2592797682710217660019399)
(-597252591996410953771036130486054161181994877/32018236242575946636019292533849, 175180495305794350784571875586200919162302600596700297894950617994092/181174097676986150696664687605077932917906393507)
(6524040067325853001658987/34151409601, -15620470877415118603010155068786214464/6311214645674401)
(6086628120412931732375095034/46386955775209, 438142492204399694621338514886675905552808/315932139232926131573)
(-3418421087065478572678/31483321, -115535880774357313819692929189976/176652914131)
(19120725754533923152418/54479161, -2575545045111528452216584723829280/402110687341)
(3022177963265842463922401859424846477043806565/11960605569359339439620022357604, -158925532537223543609743577469886295167151508114586351582809902716209/41364688007866588877468915165855023409947847192)
(-3710907119302869034089364/52263903769, 12123876901409214242490006917143569174/11948207832342397)
(434198699401599290/3721, 266333434579837268031304488/226981)
(225091243933505977906706492/84716505721, 3375084592502807010960182345137017476586/24657670871659981)
(1201381509308770910398827636278367128510980200015859060416245179167766933292/4756025957478160252570824561293553786895900221121106998989241, 39831475919274377922912363424044366641910427651654628797442817605992783880606750239677290432606560139751195844554/10372091142288945772772381056294491635284704575433767579393120013887015624160764438838240211)
(-6855227943667531903/59536, 7074650279619162380195956329/14526784)

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) = (3,-1,176/9).

last edited by Alvaro Lozano-Robledo at 2026-09-25 19:17:22 UTC · history