curve #4262

y² + xy + y = x³ − x² − 7401989477151500580338654889462505997x + 7744846513613389957443865528201108186430449098712265269

a-invariants: [1, -1, 1, -7401989477151500580338654889462505997, 7744846513613389957443865528201108186430449098712265269]
c-invariants: [355295494903272027856255434694200287841, -6691547387761968921632670089301033347722758565163495897361]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 4223028627486731438810676051333745103263583487883191927128586549581983840471922130 ★ record for rank ≥ 24
discriminant (Δ): 42755099277003039928882741297185520815766552859283207267986325750279958166451293266988599175226011885568000000
Faltings height: 19.9579
naive height: 266.2980
primes of bad reduction: 2, 3, 5, 7, 11, 13, 43, 47, 53, 101, 5881, 226814744989081168657109, 3248339536530584000120383370373029222214941
regulator: 132275760788821354494241.38410798703234125375961425133133275379613219392331456022932936
submitted by: apkorte
submitted at: 2026-10-15 21:29:58 UTC
last updated: 2026-10-15 21:29:58 UTC

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

(1312635203679634927, 538905999620598182032370936)
(1610695042098043279, -34452150226681074977792392)
(5248002159767859727, 10650689232950484817831414136)
(2095626077077016719, 1198456416877553215427199608)
(1488707630702698927, -157471246004457433758513064)
(2258499626867844727, -1596138273075185181054238864)
(14614075526633389327, -54961052685390411269500393864)
(-1002781097107188593, -3762851993126835280433565064)
(1682495446074322127, 232003810122712235466127736)
(2723228993506127609143/1369, -47838371940087130545726807385192/50653)
(1609384790408121247, 26381883410745417775096616)
(1528565309485370677, -43968414426042796319733364)
(1471378285174580303, -197955347484052125740904392)
(908576448021255457, 1330268094875518231309705106)
(15585474671147348527, 60648186658869770406744869336)
(943893790519400527, 1264556476444003641157942136)
(1689083145973349647, -247443371956905336018409864)
(1609014410787270007, 23669173735396903410893936)
(1846583780015794056847/961, 23391127706546805029693737642376/29791)
(1532927567793358387, -17566511609873756010852424)
(6751145653514428687, 16293447622047650076302076536)
(488029372155137047, -2061235914459270979570635184)
(-886061278733770978913/1681, -233722612363883599846106344460744/68921)
(1469934113376424909, -201259957567302960716732218)

Commentary

Specialization at T = -1370/199 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -4765105671341170713012T^8 + 114715794415427693698302T^7 - 1289703173484205490556069T^6 + 1970946774680673720492576T^5 - 1710136570699445082168780T^4 - 705937567588433468644896T^3 + 1908320361802866590386896T^2 + 127404060867858778875648T - 648856125950076308077632, A6 = 96897205978949562584426677136484T^12 - 5182597562294583468083790262057896T^11 + 84877162983356922665937799396845291T^10 - 478870362976518748392317900380853010T^9 + 1861233683275843787028761356431736485T^8 - 1364843279709253439363789384039551872T^7 - 1311792976812619745745993240617168352T^6 + 735145326496711370457296224143481152T^5 + 1094740667609718961081479717092760480T^4 + 53782848467549070290540339428638720T^3 - 759905051149062814344760081603028736T^2 - 65476590517227188924250720852312576T + 200491211946380337902432377627289856. It is a 2-neighbour of the fibration given in curve#3900: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-15 21:29:58 UTC · history