curve #6165

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y² + xy + y = x³ + x² − 19652590406185521977167944370620x + 7802153673263895984694874651981095905397328157

a-invariants

[1, 1, 1, -19652590406185521977167944370620, 7802153673263895984694874651981095905397328157]

c-invariants

[943324339496905054904061329789761, -6741060773700013205708917926099578642723264951009]

rank (lower bound)

≥ 23

torsion subgroup

trivial

conductor (N)

435115839382894306387158286060774650624153704748902133729273287393899976638480289172690

discriminant (Δ)

459482326388336387544839150080178031059106312214840653218112591487958375330235185366360640000000

Faltings height

17.0466

naive height

227.7809

primes of bad reduction

2, 3, 5, 11, 6963703, 54372998219597825543077138908347, 3482310011988839634083087533398258426977752873

regulator

2245649124894785075498013.19979385712290550045513899907524907069895561231664510985239

submitted by

apkorte

submitted at

2026-10-30 03:52:31 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-30 03:52:55 UTC

Witness: 23 independent points log in to add more points →

(-86955941041203, 97521308806595417816601)
(4349386440176247, -67849048604862448366299)
(-854639642230883, -154834750486332054072439)
(112606844561197, -74770083437981974888999)
(392164979933647, -12466287751760114919349)
(31454403728943797, 5523584356167484260176601)
(-27924648872453, 91383387916469585454101)
(-349720619159953, 120964033460528478055851)
(352040241624247, -30451275455271691502299)
(-339262680641703, 120126996820099014643101)
(18326290752130697, -2408854833151639367684499)
(13679238940622797, 1516130083238431306936601)
(32200029841858797, 5723751287014832554924601)
(8381096797460797, 657117534178990870000601)
(-4048155772535735, -144980977681154765755219)
(-130801937883871287/49, 69656365423473241168520463/343)
(71802636578892205, 19203741902460347865787001)
(-3419997766371203, -187115896444120892365399)
(-449655341359705200723/108241, -4744751464129092999523462724511/35611289)
(-1978462312691923, -197331552861773086917159)
(-16397744211203, -90135494746686501853399)
(-378831283835283, -123259086533810995575079)
(4320365268782797, -59479450260772575799399)

Commentary

Specialization at T = 48/85 of a rank-17 fibration of X1008 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -2350252713284250372T^8 - 50405464050571118556T^7 - 952448555730502901313T^6 - 695049490520215525224T^5 - 2931749924460790132683T^4 - 2386411337210880522384T^3 + 2409436881372804430248T^2 + 4544024515422355802304T - 2684309556378273707952, A6 = 1416965056476304965893460864T^12 + 45023054173800546171881279808T^11 + 917608982150765091196202404680T^10 + 4283108867178449968605589545480T^9 + 53149163760305922465025229930778T^8 + 64588783908002903981500312561392T^7 - 30251100417436562854623526202130T^6 - 127076856645436090494117187739472T^5 + 56548211197412673092440132446600T^4 + 156704525825325938035598980257408T^3 - 11739117611564936890190411272416T^2 - 136493880790478548113078471588096T + 53534237645294085780370164274560. It is a 2-neighbour of the fibration given in curve#3589: 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 2026-10-30 03:52:31 UTC · history