curve #6007

y² + y = x³ + x² − 49003241804398016478761406848884600x + 4169056313281125281588538758452656159915828534512656

a-invariants: [0, 1, 1, -49003241804398016478761406848884600, 4169056313281125281588538758452656159915828534512656]
c-invariants: [2352155606611104790980547528746460816, -3602064654674892257405431126969723668050561026297699864]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 1652702000228020271965360611404321303811592321821438740333392388568027778059779929380885
discriminant (Δ): 22425350057107283047258180304100485879309103583510003340500726335550819636762625722983813442066104703125
Faltings height: 18.7253
naive height: 251.2452
primes of bad reduction: 3, 5, 13, 19, 23, 29, 103, 32797, 39778113426263, 6892286526747529, 1915403630441793766415352356899227989701619
regulator: 4133235204131110976155240.9999634305148840516836220861120517709462011127448261580432633
submitted by: apkorte
submitted at: 2026-10-27 22:26:22 UTC
last updated: 2026-10-27 22:26:22 UTC

Witness: 24 independent points log in to add more points →

(98218300488716000, -17422309821030071079995888)
(123180501781478405, 1372538335598727018196117)
(23213118431479175, 55172879391262953395407312)
(138364261300018550, -6139781491270662906320813)
(115939526858147450, 6789678680812637818703587)
(23283769808497207325/196, -13493922257504603846912360647/2744)
(212657741943677450, -58010565061050764255250788)
(77322724722129800, -29022224827367874851416688)
(132177633064057874, 1090658254284491732509027)
(123725908678302155, -307139483490693353080133)
(554749342756735160, -384326545572769002165444848)
(-254355658776680050, 13315805906605814863920787)
(-68050175609157062, 84785653112558675069138716)
(368732203277632445, -190352517597875449779589283)
(123710339197415738, 377309075563326695665891)
(38914847927082200, 48177109806994695021602287)
(103312868620106513, 14460490240474793528133958)
(147426296313726200, 12203630508261713825858287)
(122016263074287950, 2537350070962839545981587)
(261693042571388705, 96264419920788249854485417)
(-113262496489618225, -90919249812252104782282088)
(6550354887336751075/49, -921234517613216302246165484/343)
(8812465984853606825/64, 2907079559246467567787442869/512)
(806080548403156355, -698879180833120940275685693)

Commentary

Specialization at T = -65/21 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -42783457648311062145792T^8 + 356652061732127265219072T^7 - 776392718276863895491584T^6 - 594563894687761097733504T^5 + 1610134904768785320105720T^4 - 1299142963618345803173436T^3 - 1310847137770922715048759T^2 + 2293995846725083806581718T - 762637489081909275352167, A6 = 3393863243684538700888419573325824T^12 - 42546028233387428372499985647230976T^11 + 184326642514085964348103427989426176T^10 - 202262449736636480735819511193036800T^9 - 605066230051264992235683353165205120T^8 + 1246062737155179656440992962298863808T^7 + 1477591518827520114475641040966728528T^6 - 1553487810870706704105291467455269168T^5 - 2067093448474470395926495237989292755T^4 + 1400402544246467407554684971457155760T^3 + 1384929818086858833031645353525974259T^2 - 1332090731990002399128362594486258406T + 294374686993228956276286908014904426. 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 2026-10-27 22:26:23 UTC · history