curve #2761

y² + xy = x³ − 342749385173046190306057584602817825235x + 2442363273510715747841541058367325989435284665192578072225

a-invariants: [1, 0, 0, -342749385173046190306057584602817825235, 2442363273510715747841541058367325989435284665192578072225]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 37185313994818665091723230
discriminant (Δ): 34171746473723442109680710636384737061680382132567423897311151405873137186047953448726281188570414583110041600000
Faltings height: 20.7675
naive height: 277.8038
primes of bad reduction: 2, 3, 5, 7, 11, 29, 83, 181, 2099, 2733361, 6440141
regulator: 3729197.352641103605736200146936132011872683790
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:21:15 UTC
last updated: 2026-09-26 20:21:15 UTC

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

(21943666966226335359023928537489074711144265380990/2039392943780604606949287660241, 1114401657192325968475710505155899096250597110939777171105887795510821535/2912402321042539817372943120380299346313992039)
(54584142614285135053022030/5987809, 127249722279607641449760207776294816695/14652168623)
(783616939307584688431305442845144698344461470/73549359416310572290455961, 93270592876920580079188440531922713703537807031828520601737595435/630766096153050653801913902600105531459)
(15698463104715642008855568086035234630735598/1462776769056359744083009, -369862139878987377120005300361202857916558107935100381148074983/1769159878565794507506804362510279777)

Commentary

2-isogenous image of D-z10-005, rational kernel x=-85689871598856534129/4. 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 20:21:16 UTC · history