curve #394

y² + xy = x³ − 354803089674674467206754048738095x + 2558194545892203175112161719607326368645810580537

a-invariants: [1, 0, 0, -354803089674674467206754048738095, 2558194545892203175112161719607326368645810580537]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 1593562111507190066539814084004447718921281851572777685020200143306222910 ★ record for rank ≥ 21
naive height: 236.4610
Faltings height: 17.5449
discriminant (Δ): 31362810004008297563332830974145357962340308661810559854716023966067971553607881810277592152000000
primes of bad reduction: 2, 3, 5, 7, 13, 23, 29, 89, 43207, 226549, 22823593909227592035983291, 44013936637595415741483513793
regulator: 351640015835560137294.39337380230310483256740118707430013952530013327207162781358
submitted by: David Renshaw
submitted at: 2026-08-27 23:04:27 UTC
last updated: 2026-08-27 23:04:27 UTC

Witness: 21 independent points

(-21518735422155486, -478256579137186088822007)
(-16162051516824546, -2017627317695268505559667)
(1038924778080097686/49, -733011743810123946757418901/343)
(-14545633423847586, -2154421321999041799859307)
(-10943447218123569/4, 14984597983246077098605569/8)
(10125440811318864, 61344439585521052023093)
(789701034488124, -1509469661698186222233897)
(228385634227786349/16, -40580603327174753093901823/64)
(-21627184418104416, 340283723703987370480203)
(8007779902166814, -480109043182118745601707)
(12800689418722964, 337581497100471707837693)
(9953363946011664, 113068979322481106243643)
(11790929866630764, 118246915756400740772493)
(945820846806594, -1491127257554750266981227)
(16992684482716974, -1198247797994777274230547)
(20859855244894382, -2057638595340369190119955)
(13765446221801214, 531556029601305747519693)
(-2970309185321538, -1893637547361814114859787)
(-21725671106393538, 109128648895082432309325)
(11620482431468214, -66260339865015976889907)
(67111185729163791/4, 9219933694407166168945929/8)

Commentary

Rank-21 conductor record from Elkies's rank-17 K3 fibration (arXiv:2608.25406), specialized at t=3/8. Four extra independent directions were found by an exhaustive search of the 1,311 norm-4 half-lattice quartic covers using ratpoints; independence and conductor are supplied for exact server verification.

last edited by David Renshaw at 2026-08-27 23:04:27 UTC · history