curve #2723

y² + xy = x³ − 202138198490462147104346858546829375275273514665x + 34979269837140650385384494084551028931706588203982251287822589358438617

a-invariants: [1, 0, 0, -202138198490462147104346858546829375275273514665, 34979269837140650385384494084551028931706588203982251287822589358438617]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 15193969795647382423209109974390
discriminant (Δ): 24245206327522789654784127343915552622546704716100437407309122024840280691220119820777537519461675437386633088780791861889526097762713600000
Faltings height: 25.8507
naive height: 338.3894
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 37, 47, 83, 269, 317, 571, 641, 659, 390001
regulator: 6814541.865516143539185602754435532272718597064
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:19:24 UTC
last updated: 2026-09-26 20:19:24 UTC

Witness: 4 independent points log in to add more points →

(47562413266946162062040056049702208330738778/66103956816964716169, 275113579858945331488197914169361058897711278158195980477729502529/537453857179830266520246035803)
(13440736921603262145027838530754623747802/47666575442058169, -208429144591835735641819999183553392457494768894867453258663/10406889119559854095476947)
(123782311552578192845827408894869379103457408603502/34381240809632044844408281, 1366910733929288893445275178859908487291269215490034667063462487793539624725/201596189655664066874438584074890990179)
(249308336890730379911847604803754614075980087777137/304911686631524918886666496, 3435404984582413523921674498343176149515544357895800994116043665857860219505/5324282682420116454602245077481602707456)

Commentary

2-isogenous image of D-z10-023, rational kernel x=-2068478264036700474688465/4. Source: https://dujella.github.io/tors/z10.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 20:19:24 UTC · history