curve #3524

y² + xy + y = x³ + x² − 31112173645928650795818908601915x + 779515812257087396130143737786847333166985805001

a-invariants: [1, 1, 1, -31112173645928650795818908601915, 779515812257087396130143737786847333166985805001]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 1789399932124511774089607156478992325365460369832460586
discriminant (Δ): -260575201105740751487391860696010282171113275960124319765544701277019673067197498463302599673420288
Faltings height: 17.5639
naive height: 234.0731
primes of bad reduction: 2, 3, 7, 11, 23, 43, 53, 71, 101, 107, 113, 193, 283, 491, 509, 1361, 2287, 4057, 6899, 23857, 49157, 611071
regulator: 3465258965864872.003547908546988461646219807766022946825934367065280905
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-27 19:17:23 UTC
last updated: 2026-09-27 19:17:23 UTC

Witness: 16 independent points log in to add more points →

(-1682550622615141, -909450579431753561532906)
(11103423640239779, 1342743841352567747945574)
(12011273923554162286027898280534631/1335397569105205089, -1709620633731690371275947158347993477629043038322198/1543177462540680150089633937)
(541109145531615611385249291/39835768921, -13453546410506579946935553759122076567666/7950781283173469)
(28310383025192151, 4752774059655245831900366)
(291845636719960919574659/5938969, 157164407907021767306006828684586486/14473267453)
(2838220085186999, -845030106720208270088946)
(88430199661220391, 26259162112520526606325806)
(722205914989991511/49, -643777446070054253059977498/343)
(2564637586401003230919/429025, -252464417191073420713520159825482/281011375)
(-69194248789413704486732027123721541084848896286383/10756494622902445717925399773845289, -942303949952239829960964899470621535352017815179013602922996874000353216208/1115593940535946065843524812688882185574973976021163)
(-3687447116938960334013927893/489987400081, -262881941367300386739292132167595035031846/342986770170099271)
(2459295138534263198341/4225, -121954237458136796238155209617506/274625)
(870151985932050521145047784159892562271/4012267223776094534161, -25660528201467872446821985505852120057553418133986530110558/254146875653888517356956526494841)
(13926388019857902457145813803037835/51678284226285253156, 326236011095895309261223172843478122066489900188891167/371502832435772939069281108696)
(23490564532034847, -3607065627114951000813826)

Commentary

Rank exactly 16, torsion Z/2Z. Specialization at T = −1247/882 of the rank-9 fibration of X400 with a 2-torsion point (Li, 2026), and its 2-isogeny pair.

last edited by Alvaro Lozano-Robledo at 2026-09-27 19:17:23 UTC · history