curve #4608

y² + xy + y = x³ − x² − 17392039633603785272776962858845937x + 913431467296259790598681781396360712331881723884961

a-invariants: [1, -1, 1, -17392039633603785272776962858845937, 913431467296259790598681781396360712331881723884961]
c-invariants: [834817902412981693093294217224604961, -789204787743968455320580498268038036534921831925884209]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 10258814917561359743040473223746249383896581362469061792611616516072656890
discriminant (Δ): -23751231429248518048769053899687622491790635613026186809384497877981425273238270692927098072958976000000
Faltings height: 18.6035
naive height: 248.2057
primes of bad reduction: 2, 5, 7, 11, 13, 17, 19, 23, 29, 167, 2539, 131591, 85256873355920076737297402339577182685267751731301363
regulator: 213466848627958616658.6025584842471602631937550697399896368366783539756445729624609
submitted by: apkorte
submitted at: 2026-10-19 01:37:46 UTC
last updated: 2026-10-19 01:37:46 UTC

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

(-102144673279320569, -40301435695312980551427116)
(-127780951948521769, -32394429754126736260628716)
(178814464393046871, -59338069280910418599516396)
(60033084062719871, -9256953896353571096032396)
(64261330401243351, -7820755765081802336373996)
(148610896593602931, -40135844741147137805579116)
(452392851507324831, 292629112820022919897092084)
(70780978395535511, -6083981397429245176290476)
(28786475679849991, -20895697569749772513928236)
(16957484969833511, 24967627996411724611002964)
(76105889146375191, -5532470704952123741312556)
(66677924433965591, -7086098258982298101874156)
(102604780328954391, -14461037648188773980245036)
(-7430079495110169, -32283827507646480479577916)
(141682478285779919, -35963976260220376189536844)
(71555340291363031, -5942510048594291931022316)
(25126203285879799, 22187797796650717038495636)
(70007413694822581, -6242495251366420107789416)
(687136370854856311, -559820112746345990675732076)
(47387628705545231, -13988459927485690855672516)
(-150439318405840361, -11186470681110855932265516)
(38933644107196711, -17184665242725156717672876)

Commentary

Specialization at T = 175/4 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -951794051889111932592T^8 + 44931125401781237315136T^7 - 935996917001646313472256T^6 + 350884544094515993044032T^5 - 1398456319949084806406280T^4 + 1757156899789831824711432T^3 + 1161615213018494249920164T^2 - 1209850628288081766896304T - 608061871140582830645232, A6 = 16573004223094294578838161979776T^12 - 853911819873454385971944861242112T^11 + 10730626760581365984133349196446976T^10 + 28681501792522102389650863794343680T^9 + 1144251410928840595115561345197638240T^8 - 1598280314283758871148427065729397664T^7 - 426571032276915684981278470103405232T^6 + 1752000471251825933878865739300711456T^5 + 357646791216195149324748670252742580T^4 - 1462595841433547195589341024424494400T^3 - 249356805352454598151173826415660496T^2 + 545782915696199470675908824491330752T + 182530075488483105810112141784098944. It is a 2-neighbour of another fibration built here: 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 2026-10-19 01:37:47 UTC · history