curve #451

y² + xy = x³ − 125314579511982544183702370268945x + 515796522496291694328940607345967092483061434025

a-invariants: [1, 0, 0, -125314579511982544183702370268945, 515796522496291694328940607345967092483061434025]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 19743833636741367656282934696071380406607013504205890312077895145804538615210
discriminant (Δ): 11014220843264556546447448734799935278705945925559546130402155148854354537539562363089294950400000
Faltings height: 17.3790
naive height: 233.3387
primes of bad reduction: 2, 3, 5, 11, 13, 17, 53, 2687, 9027637, 138019411, 373691037405163133, 4082768862599049033340737349096117
regulator: 123367931048025372478.073048759299172440538915524437839107849804032657462050317
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:55:03 UTC
last updated: 2026-08-30 15:55:03 UTC

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

(10635368900033490, -621295196351031348699645)
(22943999582720010, 3117518619081170241383835)
(-3759864604925265/4, 6363684426942023702464065/8)
(8014563762558570, 162037555206235925526075)
(5001305407997940, 118987263395518717539105)
(125197160910015690, 44127182823092593448348955)
(68669164470090, -712173854392650452441445)
(-9331806730053910, 934114456430677340551355)
(27595969394288490, 4251233395942110140075355)
(39494525707686090, -7561138254014203156937445)
(137682676126364826, -50923928215193560374009813)
(8271383322488250, 212520014320607662366155)
(-11714068234196430, -613467765336802272443925)
(2894678309835210, 421077316002349750756635)
(127646469230866770, 45435041899537238972756355)
(7565030616363690, -27059852451236779584645)
(7562757388226250, 25038624217646470637595)
(4982155776332490, 122988696623149476995355)
(1188059119004370, 607118044401419029846275)
(-173832201139235190/49, 328226801747061958228499565/343)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -10/7. 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.

last edited by Bhavik Mehta at 2026-08-30 15:55:03 UTC · history