curve #4265

y² + xy = x³ − 3652898377014660701172803041757871963x + 2692235579104960429687437813918897222945718176444904017

a-invariants: [1, 0, 0, -3652898377014660701172803041757871963, 2692235579104960429687437813918897222945718176444904017]
c-invariants: [175339122096703713656294546004377854225, -2326091540346685811512954954370982771109542323454963852025]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 187379184918627485277279847733750785346305832188682174179917091245277597430578005994654650
discriminant (Δ): -11637497363592017416985934095128842755189390620210075178712696919847364014368917553289946253638013508152320000
Faltings height: 19.8118
naive height: 264.1831
primes of bad reduction: 2, 3, 5, 11, 13, 19, 59, 67, 151, 6558883, 28798873, 63900859349050932242093, 63815270023316305769399368949867003161163
regulator: 22748363104840810922559183.60294860684563129478819257455958074103236567929151490423649369
submitted by: apkorte
submitted at: 2026-10-15 22:28:44 UTC
last updated: 2026-10-15 22:28:44 UTC

Witness: 25 independent points log in to add more points →

(1142049976083013158, 99969507698260461096168225)
(1140330748831843862, 97760600640963336247918529)
(-1996669794612390858, -1423293149148674222638709631)
(1077445381045048022, -85035878481432890267080471)
(1748562572028066252, 1284948898000132622168234259)
(-1906736959137543768, 1650797276690147569742755449)
(49609452355003470222, 349163552139969950590684480929)
(4865032674700084083/4, 1762893178284799749413357247/8)
(855143728894221102, 440254557695811466118938689)
(1083623427640235502, -79391456333220306071414511)
(5707984858373401182, 12954301285515584673754284849)
(1256816144890565838, -294047859737673631409794335)
(26825074541518692744/25, 11215951541167473182753543637/125)
(1099468843349943942, -71136895931690856578861031)
(-612448414140890034, -2167884278750929484932039071)
(13439746704598563663/4, 42595503738405236673414759657/8)
(1136340994128112746, -92851564399160301811542195)
(1742293456447524702, -1271494049205345067556112591)
(12202033385612358456318/10609, 121324449684165061935076195748103/1092727)
(1826434325378163222, -1453681564291119860135578071)
(1083109737263889342, 79816869484075995022166769)
(-1722665753418944034, 1967946832464107687138829105)
(567485861020060127091822/380689, 175680266743856442972329085424453785/234885113)
(1049141000299031262, 120897906950414320000249329)
(-944862567919172532, 2302212421659292279695823905)

Commentary

Specialization at T = -765/779 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -43748288271748043306667T^8 + 231005568889113088943880T^7 - 41811485832351502958160T^6 - 845807620575605275473120T^5 - 1273696848822391714464840T^4 - 2479538071066771770038640T^3 - 723198327599392266394800T^2 + 1473487148401141809248640T - 304202213366895598996800, A6 = 3496542619563277295766359856805926T^12 - 28800862572418754868158883533967960T^11 + 61983572330491650413223232787176320T^10 + 92074250730656596945854018024980640T^9 - 554403246688053294582105340433199720T^8 + 21665370800800153534189468438161072T^7 + 2696996106708736152461893481712643920T^6 + 2003709311601233451889158133572553920T^5 - 1216187330113512252280500336077191920T^4 - 278472048982945948218206647760751360T^3 + 596780792582732967834659478701644416T^2 - 397452607403627859574187730447221760T + 82758424296652691415712452035078400. 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 22:28:44 UTC · history