curve #2782

y² + xy = x³ − 265244220311774783700959398236648427690x − 1023237209527301143079276008512559904813713796010727085404

a-invariants: [1, 0, 0, -265244220311774783700959398236648427690, -1023237209527301143079276008512559904813713796010727085404]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 6284587191562249335065454
discriminant (Δ): 742001691737077980814500910427439852570838488588001297225479464066886644323752102738739537264581635137468346647974912
Faltings height: 21.1261
naive height: 277.0347
primes of bad reduction: 2, 3, 7, 11, 19, 41, 61, 179, 337, 649849, 7302509
regulator: 2892478.507967348808151991154822154412702966770
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:22:13 UTC
last updated: 2026-09-26 20:22:13 UTC

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

(7085448708090433931272044769943119774886/12406243590199543681, -596174217467169905754008935655820609172861122627772404486074/43697888892675432378697620671)
(-725360691080955975489626553603344560778974/112114119122601802215025, -771192105116046496544144901983004428527724322717952968667955716/37539671123951744228967170401456375)
(-14116118448043896869115616037581597738756/1392965478007019169289, -1298695113386089151656149053775743538784060494331114144269194/51988888211617723988013627716837)
(-5736790445530634923095116212430541547064506/460751281989727522795369, -5843378270241171264973743455023364952837188777360856399170959042/312751807568090406804067126784211403)

Commentary

2-isogenous image of D-z10-006, rational kernel x=6914743098109976052. 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 20:22:13 UTC · history