curve #3112

y² + xy = x³ − 102812364812633731850605646208675231977647462100x + 12167508463723925912278977317426206869256513580773249167860386181829375

a-invariants: [1, 0, 0, -102812364812633731850605646208675231977647462100, 12167508463723925912278977317426206869256513580773249167860386181829375]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 14131461780078584529422651011842782186617799735577501195
discriminant (Δ): 5596175186400419759349201674257872288522340353640313517014047871833701468620630841833224333851380824334001136257938515163216374679605388330625
Faltings height: 25.9601
naive height: 336.3613
primes of bad reduction: 3, 5, 7, 11, 29, 53, 103, 181, 193, 503, 641, 647, 1447, 1559, 2251, 2663, 7949, 14341, 298819, 23023237
regulator: 35240276001065.4173206648436641456526049036452051012597899122
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:37:52 UTC
last updated: 2026-09-26 20:37:52 UTC

Witness: 11 independent points log in to add more points →

(141257414375475714584782, 21519796839361935785539848033575341)
(199036391321563943854155442945/1721344, 96490065333677405391961299022603551642993775/2258403328)
(3641575186809420872229655, -6923073843033886379901253772520558950)
(59376118330172978169143076313/9, -14468306595445507319951979509322114685015868/27)
(25398274752370876794399209499347725/104662602256, -1315068399433181334847839333306753772395150705715675/33860026431452096)
(1074843920241555035365071550/4489, 10750074646433837886643670986457295796375/300763)
(173453390832566308708898447650/452929, 51872071272199508638145544338786387805477775/304821217)
(493026883703641677487034060130706/364771801, -10662489716302206755337357934797453122180508147227/6966776627299)
(4122161988761383787263285867120207/18730933321, -36241316960588384732379720525449134731077010005813/2563534265245381)
(4244294530197556670751328090871010/185810137249, 7941497812805994987904523663716002524052240377135055/80094760332142193)
(209521965322266556645612688845/22801, -95848030820950229732576733720455579267007200/3442951)

Commentary

2-isogenous image of D-z4old9101112-012, rational kernel x=184273670355417640439200. Source: https://dujella.github.io/tors/z4old9101112.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:37:52 UTC · history