curve #447

y² + xy = x³ − 8223603166187780927160926046820690x + 289423414197390060483587925043989774858424061454596

a-invariants: [1, 0, 0, -8223603166187780927160926046820690, 289423414197390060483587925043989774858424061454596]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 45489840601228627635532811975031132007507587327025647851680639923793222795626
discriminant (Δ): -593725561778098105732039277778467901025058153749125977677261165119338828075776800959848024867639001088
Faltings height: 18.3489
naive height: 245.9071
primes of bad reduction: 2, 3, 7, 11, 13, 17, 19, 31, 59, 45856001, 22512999527441173827379, 1774129343087441674531448761910137421
regulator: 99849115668872990442.0823918198327769762141903496892247407867405684592549042677
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:55:00 UTC
last updated: 2026-08-30 15:55:00 UTC

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

(-3050623824859516, 17733644407129010447240606)
(262881884507766164, -127650050389491079753048930)
(-25654743467311864, 21988923757403296158538526)
(30953518003180868, -8033128844152082938161442)
(23387213457060548, 10482755486768994392725214)
(43367819937262916, 3787937729899385766497246)
(67212783406976414, 6350576331718173727410320)
(-39621291106202074, -23517096649218744193483864)
(128712997638918788, 36923320941468551062354334)
(48963249048566948, 2038029829301716521456638)
(93994325290766996, 18624848060331895229433038)
(127970250866838020, -36506670199943912043237346)
(34238219544710276, -6928046696869381715646562)
(929069325871188116, 891399865794604667962183838)
(82576123193497028, 13168961426818025420257118)
(136970807713942388, -41626187335393524094640866)
(40246811804290550, 4862265587396403209484914)
(35120114870189444, -6627779821615801625303074)
(153500403738398290148/121, 1897067452977640428877137904954/1331)
(-101874965787498172, 8360172975611561262077918)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 12/5. 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 15:55:00 UTC · history