curve #3007

y² = x³ + x² − 453211679137568951700443241313500x + 3713898450132687573296510423328727753572576130500

a-invariants: [0, 1, 0, -453211679137568951700443241313500, 3713898450132687573296510423328727753572576130500]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 31602563084617300276824095632860
discriminant (Δ): -830602544155307664850951325201755525925566735828168023724591178422279297005169796990220008352000
Faltings height: 17.4508
naive height: 237.1955
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 29, 37, 79, 109, 127, 233, 3613, 6659, 3477917
regulator: 137327222.931258973503496267399845810787590304332957667
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:33:02 UTC
last updated: 2026-09-26 20:33:02 UTC

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

(35452259606936709960/4489, 238193593758753171102863068770/300763)
(-8585345847341832, 2640467266171239089668566)
(4703760766360005/4, 14271817830641558564049585/8)
(334628002978089624621585/190329616, 4488882382278792034513521246714431265/2625787382336)
(3750150901875524664/361, 2443195879252480573795546830/6859)
(7988290170020340, -776702678417372813228010)
(42753070067260678892092320/3362956081, -16195368564402253131898630635863309730/195021186093271)
(-10246256009443329957323721430893/629914473021409, -41172238598535378072199672013679906426949737822/15809654545595206158673)

Commentary

2-isogenous image of D-z6old678-027, rational kernel x=12291171619722000. Source: https://dujella.github.io/tors/z6old678.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 20:33:02 UTC · history