curve #2866

y² + xy = x³ − 811863559811140673027806343197372847488880x + 214421710492766960887781252991107774115646500735071183085158400

a-invariants: [1, 0, 0, -811863559811140673027806343197372847488880, 214421710492766960887781252991107774115646500735071183085158400]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 4180423832258126825781263565690
discriminant (Δ): 14385597968278853618666842530020695856872273721667081450685964667769378774686099824847948386205187865648095332748124102374560000
Faltings height: 23.1102
naive height: 301.1141
primes of bad reduction: 2, 3, 5, 7, 23, 29, 41, 43, 53, 79, 113, 401, 719, 142159, 872959
regulator: 142133616.35844052834918490223716592616119332505753
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:26:00 UTC
last updated: 2026-09-26 20:26:00 UTC

Witness: 6 independent points log in to add more points →

(17977892075943930816275997460/25090081, 95491870501532953854565837168073342154380/125676215729)
(-765111681145067451200, -19689966332238030317089874965280)
(4188113713560922499261620/961, 8397078152303966276537538645222593620/29791)
(979983780691929317216953106140/2879809, -970123021357329224681252344991792635586380540/4887035873)
(-31473760613982336671357597217560/68409971809, -396299532086048920003681011301782999148168017560/17892833356559377)
(510463383456000088131888016339710146685521571875089800/702634512788124084103400091288841, -52845280788925252811081715697313731350633463502226628375764069776420162827160280/18624911475355214357238338496065013995909147052389)

Commentary

2-isogenous to curve#1119 via the rational 2-torsion point x=505910156890389864160. Independent witness points are the images of that curve's witnesses under 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 20:26:00 UTC · history