curve #4213

y² = x³ + x² − 69531798251329103001474804428853240820x + 440331805945825026718858572381778190955418316341411643744

a-invariants: [0, 1, 0, -69531798251329103001474804428853240820, 440331805945825026718858572381778190955418316341411643744]
c-invariants: [3337526316063796944070790612584955559376, -380446680337192823105118964434239138649906168994489393551040]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 28528256290887068868408626166511695602057583892992150721316098304212624288718334294312692700962664
discriminant (Δ): -62246931462256159527948079123946250057670054353474258707783128071266665842110156793590299032164096451695314138967808
Faltings height: 20.9096
naive height: 274.3774
primes of bad reduction: 2, 3, 11, 23, 59, 73, 149, 117404958347, 360052418004907, 116546999415497796840397, 1486035063747424426354902812453456170193
regulator: 2709767040418178959994292806.847007372676059204824957589297661901879834577009768833013055
submitted by: apkorte
submitted at: 2026-10-14 02:05:05 UTC
last updated: 2026-10-14 02:05:05 UTC

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

(-1459744115271345976, -23210341214905210122871071408)
(5653961798739909935, 15097797107617150961062779762)
(8866158697127310782, 22821254968493534662184042286)
(-4159341641238658225, -25643345886875315503909459338)
(4715383166427561314, -14741387915289283297627360002)
(256594228528237698488, -4108152425674516641672982875000)
(6558398127820761788, 16322017460630314703243075520)
(4194360568451337212, 14915774700973485736476067824)
(425546690950326405146, -8776849986627394579012267368426)
(13363583060209697474, -43562346496087012106387604642)
(15467455623189646796, 55365329023526088513086458824)
(5311266752748500582, -14861303666251731957583288314)
(21695765071238691119, -95624880950299229850858779022)
(-861329766939415365886/121, -31904706171537968406786173280042/1331)
(3175204864739458622, -15860841155270850357708933294)
(36595776963029132886566/361, 6978540806724545505200747993376366/6859)
(4951500174843122246, -14745941044001192380429258074)
(7043645820058257470, 17321387607045149710505728338)
(-7188355424665278562, 23847664920302019301086818610)
(-10473263363264475208, 4444354302070502813671476216)
(-17640343433865216865426/1681, 261244345321199240016155222671698/68921)
(543048233879840572178/169, -34743416163451673102758210133322/2197)
(10605731547048823322, 29930674139187979110654812694)
(326575073057219973646940/38809, 162380378473897364382743701319025024/7645373)
(1045911968447975227019/49, -31958572350425396593214730255768/343)

Commentary

Specialization at T = -120/121 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -18808819639564736183472T^8 - 270390604164314807890368T^7 - 1064698058908765727554464T^6 - 690609682722857600525472T^5 - 1141985348588431905168240T^4 - 1276711546748486547364128T^3 + 1533178281333365507000016T^2 + 1003847548076768965363668T - 662667526757651887035567, A6 = 198461588427944563947255082869120T^12 + 5470876228138813326458542895357184T^11 + 78310570892779922090226615255906432T^10 + 585449236963957414662829259015483520T^9 + 1854487489939448014773845858584024704T^8 + 1020492248925323261478020126191388928T^7 - 2397613146941260720543566338947934208T^6 - 1992669351542649345334637022139959408T^5 + 1587582544452083911151305726515006564T^4 + 1461616308592226634206845276455690360T^3 - 684661031717307107294832767505632268T^2 - 464296295429065890603785053139616684T + 208515476578669742594974588404489810. It is a 2-neighbour of the fibration given in curve#4191 (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 02:05:05 UTC · history