curve #147

y² + xy = x³ − 304375792648473160657575x + 65348316359688219048467264095557961

a-invariants: [1, 0, 0, -304375792648473160657575, 65348316359688219048467264095557961]
rank (lower bound): ≥ 18
conductor (N): 21419202627854163602361890682263820949374863109651290731136698
naive height: 173.8532
Faltings height: 12.3566
discriminant (Δ): -40091900068111500354981590451888913785493616412246049722990279496695808
primes of bad reduction: 2, 13, 41891672862379899773, 19665375703497198008963188787543007713701
regulator: 328231786777603894.93250258653942856450752812415634945940058946359937174453
submitted by: Alexey Pozdnyakov
last updated: 2026-07-01 22:31:42

Witness: 18 independent points

(665672967286, 397122969151581061)
(528305906806, -228032571604209659)
(1621064762358, -1957512507166916987)
(141813046806, -158227285823597147)
(-449951132122, 333477709769604789)
(-309393665930, -360421436078099579)
(360905322078, 50063259539241389)
(-592171937290, -194770351490693499)
(-635714776838, 43943567005787149)
(-169705070602, -334835632751779451)
(942176398838, 784180116547849093)
(-86690933482, -301800268160975291)
(1293065099350, 1354177290109513669)
(16301299883334/25, -47452911614929710287/125)
(-1515067081250/9, 9028530617604404087/27)
(1623779416726, 1962768340928734021)
(8394886876764/25, -3961023236839620973/125)
(2206741920622934/9025, -63832311566160807668517/857375)

Commentary

Generated with assistance from OpenAI Codex, based on GPT-5. Method: generalized Mestre/Fermigier quartic-family search, sieving Mestre (u,v) families by Mestre-Nagao score, rescoring survivors, exact-height triaging the best candidates, and certifying top score/height candidates with GP famcert. This hit came from family mestre_u2_vm4, specialization T=3433/2. GP famcert certified rank >= 18. naive-height record for rank 18: h 173.853225 < 190.774340

last edited by Alexey Pozdnyakov at 2026-06-24 03:58:36 · history