curve #434

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y² + xy = x³ − 119748659652394027153495287863671115x + 15870961531086086871587619570217861268496637395066225

a-invariants

[1, 0, 0, -119748659652394027153495287863671115, 15870961531086086871587619570217861268496637395066225]

rank (lower bound)

≥ 21

torsion subgroup

trivial

conductor (N)

8326559330120298582074074068589703981740465847850520002266773544829744538239330

naive height

253.9257

Faltings height

18.9958

discriminant (Δ)

1083183267000326135418099570935273303254315599120654386377035262770393261696249926038970689754716416000000

primes of bad reduction

2, 3, 5, 7, 11, 13, 17, 23, 29, 36353, 651943, 859287078407, 171533634337430750969577639730984194402389843219

regulator

735004338681961160047.70581984974834515047498418730602798814733316858827995015145

submitted by

Bhavik Mehta

submitted at

2026-08-30 04:29:51 UTC

later contributions

rank ≥ 20 → ≥ 21 · Jason Ma · 2026-09-03 16:31:26 UTC

Witness: 21 independent points log in to add more points →

(232947900586241110, 24831759567696378492034705)
(185846676119451430, 5918369614431624233449585)
(-1500955330983244280/9, 4769403201808716117844741045/27)
(216572043063620530, 9732708348978505944970885)
(85097858335547392, -79352728908163968825248153)
(-388544912990165168, 61165154852172504772862647)
(-391440734938733210, 52598725856744853398332585)
(239108089009371930, -30141967461169985370905415)
(1484016079930725646, -1762502813953854779714571743)
(314609954198946130, -96626875443399893613847415)
(-26216775763099298, -137812793269428734988273503)
(-68822984830286690, -154228514011438618725242495)
(666725929447237930, -482085828257917450652365415)
(2270388951390146575/4, 2892891580565805360604773905/8)
(218347494902512630, -11575652813224572493162415)
(-314101840271809320, 149983195871484412405438335)
(302863134182143870, -85929940440540858266529215)
(-2019794321406410, -126936286985641471736314415)
(3042688505153830, -124525627203002953018006415)
(-4954306966644920, 128312554125616002986852335)
(-60601407603527090/9, -3486763588060286085645067525/27)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 6/11. 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 04:29:51 UTC · history