curve #242

y² + xy + y = x³ − 309526258964080213728331769x + 2161609688903376164317636000643138896976

a-invariants: [1, 0, 1, -309526258964080213728331769, 2161609688903376164317636000643138896976]
c-invariants: [14857260430275850258959924889, -1867630771212539291861082918331060446874813]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 4599105032114305107130646860970904298444214155782477625288488806310
discriminant (Δ): -120648124472674441798657772129443506216935731373216705396040170880065931985481700
Faltings height: 14.1373
naive height: 194.6665
primes of bad reduction: 2, 3, 5, 17, 19, 127, 2804141, 94314161, 935018818255279, 15112931893627108554703387672603
regulator: 663930766627720694.99921613424554549280042784189520445712829297273121330175
submitted by: Edgar Costa
submitted at: 2026-06-29 18:45:02 UTC
last updated: 2026-07-01 22:32:42 UTC

Witness: 18 independent points log in to add more points →

(-34259847583749353581038809947044/1740884375734350625, 57715002807876692686002268734914592907285522529/2296967716638686767512359375)
(7525863514713, 16075189730766472378)
(-660198343256415048/37249, -327862116505167218638695464/7189057)
(441204609390461774130/27804529, 5174310853255902398806508350454/146613281417)
(-29703234518830387/1681, 3168768482019896541111408/68921)
(67901882238708163545/4946176, 245881666667029467951608428843/11000295424)
(92226852264191646/1225, 27305643776116186645122701/42875)
(10689578885677356867/1682209, -46351324941411708197844205556/2181825073)
(-27023096575177356/3721, 14402950374183416641536268/226981)
(91792253098923102966/7187761, -329018460633611310010897992958/19270387241)
(437563141366809, -9145667001043674681158)
(96577445967179930765719144/4431913354521, -708384688894540163927332586316702566707/9330112744984508931)
(171621665403003489/12769, -29899924168825630281171868/1442897)
(10464469825449, -8276152107015445988)
(396411283986407896122/25270729, -4338197160188729826225491076166/127035954683)
(60092165892459/4, -240287988841674654337/8)
(11265307198908801/841, 499198982427503899130108/24389)
(17714909651077851/841, -1722454784822183933725492/24389)

Commentary

Rank >= 18 via 18 independent rational points with a positive-definite Neron-Tate height-pairing matrix. Mestre/Fermigier u4_v6 family, T=4694. Independently cross-verified in Sage and Magma; finite-field product independence backstop. Lower bound only; no exact-rank or upper-bound claim.

last edited by Edgar Costa at 2026-06-29 18:45:02 UTC · history