curve #430

y² + xy = x³ − 105854341112379836965356844298984005x + 13004863393089789470296160259493256256960691084729025

a-invariants: [1, 0, 0, -105854341112379836965356844298984005, 13004863393089789470296160259493256256960691084729025]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 5396999619055605101014796305076019008577308323926541422200477051902710
discriminant (Δ): 2848588085384977822706970819409814653720503217702910761115999942195145917806629357405166589414891520000000
Faltings height: 19.0233
naive height: 253.5557
primes of bad reduction: 2, 3, 5, 11, 13, 17, 29, 37, 41, 59, 89, 241, 306700111427, 762057475671611363423, 5687244150984307594789
regulator: 2375998917746104211.55565137818131554559875077824777255834185459496606969923074
submitted by: Bhavik Mehta
submitted at: 2026-08-30 01:39:36 UTC
last updated: 2026-08-30 01:39:36 UTC

Witness: 20 independent points log in to add more points →

(-59623723982707090, 138218420506792882011161795)
(230246921987828930, -28956100975926655963759705)
(165517701987591410, -4317085088165074338739705)
(347828923871655410, -135158168170787967886579705)
(150040986983772530, -22363831444324536892744825)
(211976007370422050, 9548622471470811185546855)
(3239492210957364386/25, -4782863101314740184234527381/125)
(-359969657639359090, -66820733417354976513762205)
(-2144357240355718, 115029748568770879857256151)
(-39262900706523310, -130768823316748908387194905)
(63113187121730930, -81089183630021117214511225)
(332524153188937010, -120721592345284634903165305)
(234477926584306160, -32801363944503226285433455)
(145453852512623410, 26177912679111770061300295)
(2311832550186831986, 3481958879121881284729237127)
(691979002698632690, 520672263519611379749022215)
(-250719695014802830, 154221593470920245344146695)
(49403486037160010, -88858703803957531841581705)
(1007372537880268519390/729, 31155245874568914140346382161485/19683)
(-360445158421899790, 65804684231095531883135495)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -45/7. The 3 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 >= 20. A two-descent gives matching upper and lower bounds, so the rank is exactly 20.

last edited by Bhavik Mehta at 2026-08-30 01:39:36 UTC · history