curve #4222

y² + xy = x³ − 296108993245761771418655128907858800x + 63098745523210502073156185786104112494063719776997632

a-invariants: [1, 0, 0, -296108993245761771418655128907858800, 63098745523210502073156185786104112494063719776997632]
c-invariants: [14213231675796565028095446187577222401, -54517316132053873812526792032888800737014223168691787649]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 2338316115837191674733629561727536281841554758951445057531174296600730
discriminant (Δ): -58355435365895417048106868394053496017787390947135560508459480149762425027833237290232922431669779456000000
Faltings height: 19.2784
naive height: 256.6762
primes of bad reduction: 2, 3, 5, 7, 13, 19, 23, 29, 31, 37, 47, 53, 97, 13119857, 42534538278261919263861718677607149403105997
regulator: 4894717207613552166.9120291293895279491717325424063584941956092099141176210737894
submitted by: apkorte
submitted at: 2026-10-14 11:24:57 UTC
last updated: 2026-10-14 11:24:57 UTC

Witness: 21 independent points log in to add more points →

(417863685871341104, -111034920099736374623063152)
(656654533550033504, 389619942862855830799983248)
(-629028544111057696, 21626008808152031951285648)
(259103251668985664, 61406094723086628920084528)
(-297291614797214251936/841, -8574365489872302974974192167728/24389)
(-847283388019041464/9, -8106335586195297837523180304/27)
(309363709606356254, 33185880452605700264412998)
(354276784437175904, -51577180105724384728362352)
(369142766852505404, 63984387481251209902574948)
(426899139081554534, 120371913668410483425198428)
(306850692166313144, -33611652528667191212376112)
(35759078633705376344, 213810726825527294443035568088)
(815355432752793782, -603089226875631178529649682)
(13526901258483095654, 49710866542943621871645083198)
(249068053133579984, -69270236207520892476438112)
(216398962856902064, -95680237001335437604039792)
(-220504948380516496, 343031714209958664699663248)
(-365337128810599096, 350023375804614823970986448)
(367880475830642624, 62876995903343609450149808)
(-461109156560342896, 318740524771527507103272848)
(42982957154016929775584/3721, -8901658654979885790514712213592112/226981)

Commentary

Specialization at T = -11/115 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -1294316468457527272747T^8 - 61482963914543295234088T^7 - 943435684374763790645824T^6 - 2688743270213054211837664T^5 - 3856670928366631764978280T^4 - 999902363213138391166096T^3 + 2961913384000561526356496T^2 + 1046903749166543421058688T - 943686476906321498328832, A6 = 32662063813788609918746258888614T^12 + 1796450107115792990741776388684984T^11 + 35062265377401817193324124680756816T^10 + 316874613294458685885325494699025760T^9 + 1878352364777031985219759926555560760T^8 + 3544464889310281997809284624918208080T^7 + 1303911666228891105042353858020750800T^6 - 1471880942788787576791593483327263040T^5 + 252670343673375657835921804367034960T^4 + 730639334234102885410704041977318400T^3 - 1097948334966447155670068911813786880T^2 - 383111400250142557306211553242163200T + 375132589068583102936074200430223360. It is a 2-neighbour of the fibration given in curve#4187 (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 2026-10-14 11:24:57 UTC · history