curve #3132

y² + xy + y = x³ − 3548818984538903893936299908730878473x − 1458326273185123343429141360947073979888405493321482244

a-invariants: [1, 0, 1, -3548818984538903893936299908730878473, -1458326273185123343429141360947073979888405493321482244]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 233675553798276919489685212875169920733360530
discriminant (Δ): 1941690166862869654225798673188381676791619807992217911851987485964956457376782707237134977419666950731506250000
Faltings height: 20.0528
naive height: 264.0926
primes of bad reduction: 2, 3, 5, 7, 17, 23, 29, 31, 41, 43, 59, 79, 173, 223, 257, 347, 599, 1801, 5347, 129581, 149803
regulator: 32894739751.4404415023167691765487766024757032072650091077704
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:46 UTC
last updated: 2026-09-26 20:38:46 UTC

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

(17222907174954573290755/3969, -2013650810646262819432925072389406/250047)
(41290823063588027319780280051495/19998703504081, -11306776247112790313602474641329165658504826358/89434022081918695271)
(6675779752997360380, -16503441341439942292765123933)
(-34581595466257140958078921145/79646435089, -585345554763935988405288803596423911961774/22477577971512313)
(68070121438421139150926543153980/15949344157569, -498034529030025734978978869048068886189941592931/63696305636349485247)
(373003649726212886327702050299895/91744865349316, -6295121098217837162529990482694096348841219446947/878764797998082305864)
(38240704903715296033870/17161, -2929967326986743473771782924055343/2248091)
(75602957101954478780450677763324282015326454983/21110186219413840566332849689, -17287189288982198565746954579775706660194818789517881699599739580955622/3067171757375431840758442033485297627680387)
(7836607467450425933761305216880/186091292689, -21915577129323475587403440964833827027651806651/80276620114058887)
(41624597374521703108125553251210281289640124539/127894142864039454148155625, -8492152492766499823963438277395212061383936333193750660655802078381062/1446358604174159220917225878004903796875)
(122649416216339362983218821260085946013282400439008255/13366954299224166477009159397277689, -41996987855376008949337293121525768257644352716104291168575156555630947348284526/1545427709392070299423927193869583988990685170566387)

Commentary

2-isogenous image of D-z4old9101112-010, rational kernel x=13031852518377921639/4. 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 20:38:47 UTC · history