curve #2754

y² + xy = x³ − 45993795937920741187765286114079872057340x + 4066125294553521775538577563888369960307865379539920194802192

a-invariants: [1, 0, 0, -45993795937920741187765286114079872057340, 4066125294553521775538577563888369960307865379539920194802192]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 37726797710029395854920110
discriminant (Δ): -915434158332696517267362167614137498678017349151859582227108000488958795888095204992928847078459696123735814195154124800000
Faltings height: 22.3390
naive height: 292.6387
primes of bad reduction: 2, 3, 5, 7, 11, 19, 41, 101, 113, 149, 541, 887, 4391, 5851
regulator: 16415492.79722560201353591284197609953813316236
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:20:51 UTC
last updated: 2026-09-26 20:20:51 UTC

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

(108354933688220816016740291078988826061436156/733889661283721852836369, 441834392501275406612080894569602914503686933307467174823315094164/628703899996584691744804782804268153)
(22625977420416732488171663201376/503953750609, -517404303736481213293916902916573419689090432516/357755255696077273)
(-53628503361736560463732592763647265564397704/2428264468593304481108929, -8521114804269786326565585562751245366472600313232602909355336251204/3783937702255046630079018008088271967)
(3119464771235694212871120454386737362285886416061937478252210519528157283239770452459437368587985358231585682243101549321598614500576769531360904/24458061934941303129386730268342672434690963540466223523658155677152766347410177468778290722206850668401897369569253695643121, 2004794258821380964975304459403535047986994139234786495522360111635806945654527039917435098245792444408462433446221310320689805850792179838334760886047744925431655344448846323196836130441203234774979700962526094674772/3825014278075369745554505633993738864559251907807942590601516676365897359751442889676485369323320429510618681908025141040335084771374979636512153329063076356809649838928997903942949764169)

Commentary

2-isogenous image of D-z10-011, rational kernel x=1996576092274320591935/4. 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 20:20:51 UTC · history