curve #4281

y² + xy + y = x³ − x² − 462073956883392438587926051647287697x + 117240873056116940628997828161687268107615922183751169

a-invariants: [1, -1, 1, -462073956883392438587926051647287697, 117240873056116940628997828161687268107615922183751169]
c-invariants: [22179549930402837052220450479069809441, -101296114320485036603646148844885032909988129610946867761]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 75658910303290538667272432081894152972706180017093521752285062478585089190
discriminant (Δ): 376121082519779690832781742503478767650138384955117580720528315657208774428663148072598636627195494400000000
Faltings height: 19.4134
naive height: 257.9767
primes of bad reduction: 2, 5, 7, 11, 13, 17, 19, 23, 31, 83, 101, 30089, 2215341147311, 4517963993642910826508666609592000581408041
regulator: 239900305773727880297.9155277137260633018666668109138020920498031534885463354105746
submitted by: apkorte
submitted at: 2026-10-16 15:24:20 UTC
last updated: 2026-10-16 15:24:20 UTC

Witness: 22 independent points log in to add more points →

(-326634504311987433, -483033645852522294780794764)
(548326315307823127, -169511980221807853756743564)
(-388141799797713523, -487971262880544209610037414)
(853930899516323727, 587661748693615516345282836)
(-466184160923234623, 480975794387208579974651686)
(233798314596538563583/729, 888045404584667157716667501788/19683)
(596533477577075127, 232112112667331613230344436)
(-741078579030254313, -229511461098930553927856524)
(211584881021702627, -170133112902847944988983064)
(2065431428560809964807/36, -93861139432197757077846264521299/216)
(-219804579407251673, 456275349736164970764845236)
(261113245321977227, 119958229234086467113874336)
(607435767609359127, 246355814855392331153720436)
(147875623014273047, -228352851182529269155303564)
(220601477114306727, -161376110788118240461791564)
(312056560696538327, -58613239373175959271143564)
(335133208445711127, 4978567459413572981560436)
(-310166610712500873, 480335014985486453766860436)
(134535055217251127, 239814030686104102060488436)
(94387974547458107, -272887447743686596542923344)
(-730311286878171273, -255311246093495279241331564)
(971400488516237516623/3249, -14121707921120878937474253731852/185193)

Commentary

Specialization at T = 400/29 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -8921224827903176421867T^8 + 94634723251967943349008T^7 - 153371039643193058472636T^6 - 641050270990816586343840T^5 - 1411956329363864664615090T^4 - 3616744002660872511746400T^3 - 276286449215617413614716T^2 + 3264390669786107213517648T - 1094619040863704280589707, A6 = 270621608798713355418362188083174T^12 - 4440334258578316878189533038482864T^11 + 26574225298655897363019125632965972T^10 - 44466546059753405576217220076774400T^9 - 266551804859294565919720379747154006T^8 + 857066304185931807663129817734860208T^7 + 2882807104710383577362230811059875192T^6 + 490875434163168933481957024140491568T^5 - 2474513386731796364719321202224339446T^4 + 1633690238596691275986648934120443840T^3 + 1554365531979876509922013707743895892T^2 - 1882445996339364406394564636163936624T + 453429930884494956186985826996791494. 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 2026-10-16 15:24:20 UTC · history