curve #5962

y² + xy = x³ − 6894248928209385542157622322535386400x + 6252142718626614698595189176307466674922202725570348032

a-invariants: [1, 0, 0, -6894248928209385542157622322535386400, 6252142718626614698595189176307466674922202725570348032]
c-invariants: [330923948554050506023565871481698547201, -5401851308893395100082629371160726966168131962115328520449]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 78430510892679442785713283664025259544985871649589693665362745489233014673514150316589744990
discriminant (Δ): 4085475903989295462621558659747314733136184331035381704563088842102468726174857953339709898546080053065472000000
Faltings height: 20.1489
naive height: 266.0849
primes of bad reduction: 2, 3, 5, 41, 53, 59, 67121, 369913, 8050057880141222756991211, 102022282593753705251183255146754405699229590172773
regulator: 86174161825056695150155624.16918091009138693816889828459083086482478287363653999504194986
submitted by: apkorte
submitted at: 2026-10-27 00:49:57 UTC
last updated: 2026-10-27 00:49:57 UTC

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

(8923493454915604824, -25598786246245520711439943392)
(918997548495690564864/841, -3725076398432761614170825389728/24389)
(676382329600386684, 1377836995309131504598640688)
(5079058945149900084, 10112344721189684054384244528)
(1898293101520002864, 73269566385747757526105328)
(1286712153709502332104, -46155259930605895286574879766752)
(10241267737818614304, 31777145285211851162606446848)
(2046168377056296864, 843952975072551665258927328)
(114766335468446256288/49, 589443677951367798045686622240/343)
(2223591760187663904, -1384336111735695343940727552)
(2003456548781519744, 693804631101984706487984288)
(-28923791139158162655744/27889, -16325522212173044901296568112217376/4657463)
(-208126540482659987094/289, -16178333461537729126532458357926/4913)
(1908194690368816704, 211436369012972757036721248)
(-589778145012636096, -3180105761911262731392027552)
(663444324402561504, -1403642750340422813746997952)
(35582583689871050304, 211690262249912396398843978848)
(768258023023064304, -1187022416656992176429011152)
(3242137311971788104, -4240238129443844857143989352)
(707749012507138795584/361, 3573117007964719990447295983392/6859)
(-1139464579598954496, -3553651317728650321324433952)
(4234784802302659491/4, 2991766070601049932961020459/8)
(1926824540069112384, 348948760358408114156351328)
(1926011589082554624, 343975477701067924004398368)
(3241967223193169652, -4239743939066760656979879972)

Commentary

Specialization at T = 253/2 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -7016388255537308794800T^8 - 178679712000491349877440T^7 - 1393427674620775645747200T^6 - 570182635321524575663040T^5 - 753965899387475415418440T^4 - 320804658906630906472920T^3 + 1134421125068367717675300T^2 + 399978175235844546966960T - 422082632478729918479472, A6 = 219688715864979146315449005782400T^12 + 6098200250797428195163726469525760T^11 + 54477207303588087906057214569267456T^10 + 264783925318167622977267944863123200T^9 + 1702978825713300190413814657041122400T^8 + 932916852969092429403286366085219040T^7 - 1775033404268099174843557641423862320T^6 - 1248513743927587153502730303378288864T^5 + 1060720915885228836685362726021218580T^4 + 675987962412402587067553913812781760T^3 - 432422499228520607079300144749998800T^2 - 178229177722910279472469086969305280T + 105716686434811400528878525349249664. It is a 2-neighbour of another fibration built here (itself a 2-neighbour of the fibration 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 2026-10-27 00:49:57 UTC · history