curve #432

y² + xy = x³ − 134422478444183165897105480803299200x + 19105637503606867472088832718752567503960757778399232

a-invariants: [1, 0, 0, -134422478444183165897105480803299200, 19105637503606867472088832718752567503960757778399232]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 2021144615126579179797246594395487085729173432156567123503881134380779681747410
discriminant (Δ): -2239196697651177772152759233975931016330047275115124817633287595245219412904559503368757740795534131200000
Faltings height: 19.0411
naive height: 254.2868
primes of bad reduction: 2, 3, 5, 7, 11, 13, 31, 47, 577, 1997, 4967, 13967, 577871576750005687906572281720218749111626260720627867331
regulator: 825016574380675822272.17032094673594973504181423338175769897752097977015566293471
submitted by: Bhavik Mehta
submitted at: 2026-08-30 04:29:49 UTC
last updated: 2026-08-30 04:29:49 UTC

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

(613139836370682904, -408888655747451285936040152)
(551746626861539392, 336011296136689273549138048)
(141721995836368384, 53865753872102516777269888)
(921663449859373504, 823488205810811963542810048)
(241294366481789904, -26816519283745016181466752)
(-374544124485767096, 130040381533619639253632248)
(677904270939156544, 489401491834969236189275968)
(227412328984290304, 17240886288000551091576448)
(26365323367731904, -124819360394699452982610752)
(-75967341146435456, -169938063534571885141980032)
(410074257731562304, -181496265704046836369274752)
(248811965003423104, -32603939042290837423713152)
(21824084168107504, -127210002083015654771879552)
(255785813086079104, -38176104171018025013293952)
(267888791026421848, 48168888046161787573326016)
(-49502328603666656, 160120450656397462249948288)
(212349014807430784, -11679919500181940888150912)
(268856937647001844, 48982554997139201244521788)
(304892167001311744, 80397878558931139243938688)
(22066522863279322984, 103643313473419608478135515328)
(-250731183859414976, -192476092738925963056640192)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 32/3. The 4 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 21.

last edited by Bhavik Mehta at 2026-08-30 04:29:49 UTC · history