curve #433

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y² = x³ − 1365250332763869332711550077906847x + 20124145234049359349449557795949681874466874385250

a-invariants

[0, 0, 0, -1365250332763869332711550077906847, 20124145234049359349449557795949681874466874385250]

c-invariants

[65532015972665727970154403739528656, -17387261482218646477924417935700525139539379468856000]

rank (lower bound)

≥ 22

torsion subgroup

trivial

conductor (N)

8767703080908304246350044139590743239251700950560208898265331557899477239062701912

discriminant (Δ)

-12090981377961386591784512998993507250890722176967802591484974773130916656682489069003465618447324928

Faltings height

17.9699

naive height

240.5752

primes of bad reduction

2, 3, 7, 11, 13, 127, 173, 467, 79490093273809, 542758304000375563517, 39258620240424693354641956745935193

regulator

35622604278554946591845.62375391998939925032321571319415160914222824511689920758636

submitted by

Bhavik Mehta

submitted at

2026-08-30 04:29:50 UTC

later contributions

rank ≥ 21 → ≥ 22 · Jason Ma · 2026-09-03 16:27:57 UTC

Witness: 22 independent points log in to add more points →

(-21318875892564579, -6288118673751306392182182)
(-34850092549290501, -5037539017053598096490064)
(617722278644772973, 484652193043798424920981706)
(-35473260775841217, -4890416087340832778874144)
(34182399082140253, 3660132931879920140478106)
(17624526014930493, -1239699373965004289873094)
(36685006021669677, -4405704323526917830677258)
(-94636919450041647/4, 50076067639487289230319489/8)
(7520154055190644539, -20622194291944254695782352256)
(-169341729552119439/4, 11441078481091034428205553/8)
(50402243117358573, -8908065134347878285118794)
(162128482797677925, 63721502749831694280328530)
(3832411807696367217/4, -7497043568525118274478788977/8)
(105157259270017626, 32239607726655472612867398)
(-10824292539002271, 5799463996365775437527826)
(29435847740396769, -2332830417830666708708754)
(23421592852483497, -998105548607675724173958)
(-32834128279281, 4490987864959205492979096)
(-228290194493781, -4520598035700883303263096)
(-8893453237285847/144, -7768020799173978087621014699/1728)
(1207041375047809/4, 35518733583190143267304169/8)
(348063743694325, -4432718495404680600571010)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = -11/10. The 4 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 >= 21.

last edited by Bhavik Mehta at 2026-08-30 04:29:50 UTC · history