curve #437

y² + xy + y = x³ − 20461874431447421784527688374633158x + 1127554729426977874077743817330162797017594385944556

a-invariants: [1, 0, 1, -20461874431447421784527688374633158, 1127554729426977874077743817330162797017594385944556]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 279764396370622643502518105435046713192665829643494765937578943117437538847250370
discriminant (Δ): -938574726530719210755177966086236778183778997655149161845867777826842521457027328930277609978004062500
Faltings height: 18.4866
naive height: 248.6269
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 73, 4271, 40787, 24936695890063, 1546216870694055870470428898192667023018326472784167
regulator: 759600548686439624124.570876383637238779487807698203757503091425997693324246277
submitted by: Bhavik Mehta
submitted at: 2026-08-30 04:29:52 UTC
last updated: 2026-08-30 04:29:52 UTC

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

(13863232234353565, -29095555953791513662777333)
(75406215960001990, 3656404757241931468422617)
(-140051058154694195, -35302387679782265954280973)
(79093802549923840, 1986107876952001789439267)
(-7297588089318035, 35727978242387764389426767)
(82220130606620251, -998572533162232674544021)
(63923456120540365, -8986965612537104984135233)
(96017927653321165, -6933953245508911952874433)
(29371964736498378715/361, -7920192142681787362838972797/6859)
(-30162009486628385, 41440151897520866694138617)
(811720568109156340, -720661518287744113009720108)
(-55455492923536643, 45735491018870519710718843)
(62170130145264490, 9784306914266307624081992)
(1579021533230815, -33094545262479709029624583)
(50064133811228215, 15120526774068208521288767)
(413046305928739735/4, -86052141975137818579582739/8)
(65207523540886075, 8399424854825346272676857)
(-142932803509844285, -33647305515640605454616983)
(85076111264444440, -1585627019974099665793108)
(-150540057123058925, 28218809655020245864203107)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 33/4. 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 04:29:53 UTC · history