curve #3038

y² = x³ + x² − 24177745621979422976624024x − 33617660927432101946278517493176845824

a-invariants: [0, 1, 0, -24177745621979422976624024, -33617660927432101946278517493176845824]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1605314370710886136520856
discriminant (Δ): 416315627569180646254422946651943649242673544861505276073071062562508969452544
Faltings height: 13.6022
naive height: 186.9560
primes of bad reduction: 2, 3, 7, 13, 29, 31, 47, 151, 181, 313, 661, 1009, 3049
regulator: 12754728.1889733735203215859888936732113911641919580549
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:34:25 UTC
last updated: 2026-09-26 20:34:25 UTC

Witness: 8 independent points log in to add more points →

(-238325216411072/121, 3357309915766215730080/1331)
(-33871366027709/9, -54467762682562326730/27)
(-11663702100779/4, -27816349861671345975/8)
(-53912206060616/25, 364278840918621415248/125)
(-1697497574432, 1591435744721681040)
(1123168014724879797400/6974881, -37623810604445202324554722869696/18420660721)
(117359496485816927611/2739025, -1262712745944117727530500922966/4533086375)
(18512911122566561954839/189805729, -2515658069338424855569414998631596/2614953528433)

Commentary

2-isogenous image of D-z2z4old678-023, rational kernel x=1980591661228. Source: https://dujella.github.io/tors/z2z4old678.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 20:34:25 UTC · history