curve #626

y² + xy = x³ − 182451976602578656424609725499140x + 710003150253794219215652666162794189038512805392

a-invariants: [1, 0, 0, -182451976602578656424609725499140, 710003150253794219215652666162794189038512805392]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 5651319165610115296564894984823807468501395910978185811787187868498410205790
discriminant (Δ): 170936848285991366957118762195218257884358220481825423891549871859917613495299194842452393152000000
Faltings height: 17.5587
naive height: 234.4657
primes of bad reduction: 2, 3, 5, 13, 17, 19, 29, 71, 2465779087453622652131442949, 519784438179112504122441050306814600881
regulator: 1539741648619753344327.838064369512191998122834483177809266655540131523115768744816
submitted by: Roy van Rijn
submitted at: 2026-09-07 07:58:17 UTC
last updated: 2026-09-07 07:58:17 UTC

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

(-3428955858517076344/2209, -103275287379914878066351333436/103823)
(341451890018744865176/100489, 11453652699578302486870011699484/31855013)
(1421989050733709804/1681, -51403653211404431079338658392/68921)
(54938084184407822, 12510003464734650235120280)
(78266781050788904, 21583976314861037737936028)
(1576838726662055144, 1979999227161508319030537948)
(275856238766592504006794/133225, 144882186604686128205773785854751048/48627125)
(32343362505256116953036/4489, 5816704343975764908258919803727544/300763)
(451685153382923636/289, 3216654481209820295439953644/4913)
(159880614487269527391176/37393225, -20535320605070024923156727491612396/228659570875)
(-207063576767969910796/25921, -5373215212229459128185363202772/4173281)
(63431781447715124, 15632027594258343566588588)
(-14726457028992856, 450742903005235879493948)
(-25547628719315830766/16641, 2132137788731395250384172645382/2146689)
(816995985903150650526626/4489, 738464942904456142122944383416294434/300763)
(403133816812318948/9, 245103819466924941678683696/27)
(-21180060288786392556548096/1431789921, -22466095285091040253863758892269050628/54177498820719)
(-2891729672233387873/196, -1197251284734026121558242045/2744)
(-4452688329359660425432/316969, 126297560544003373654469422999060/178453547)
(-38963064251172216262504/2798929, -3479733264232670937714460369607204/4682608217)
(-12896965910595856, -958073803358837270715052)
(-4209096890749151344/529, -15666577260849429714360102164/12167)

Commentary

This curve was found as a rational specialization, at parameter \(t=3/17\), of an elliptic-curve family of generic rank 16. We found 22 independent rational points on the specialized curve, proving that its Mordell–Weil rank over \(\mathbb{Q}\) is at least 22. The exact rank is not currently known. The submitted equation is a global minimal Weierstrass model. Its conductor was computed and independently certified to be 5651319165610115296564894984823807468501395910978185811787187868498410205790. The factorization and local reduction data used in the conductor calculation, as well as exact certificates for the 22 independent points, are available in the accompanying research repository.

last edited by Roy van Rijn at 2026-09-07 07:58:17 UTC · history