curve #2950

y² + xy + y = x³ − 2803753156114296450454833212325478304786124x + 1806998436261713446671118650546390909182980742553050103837376702

a-invariants: [1, 0, 1, -2803753156114296450454833212325478304786124, 1806998436261713446671118650546390909182980742553050103837376702]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 122397238576536479651595594069449949746370
discriminant (Δ): 332280796251037410218221268922713396876335331802926593035799574817180361624915425115218891019736585154207389016544730940
Faltings height: 22.7895
naive height: 304.8322
primes of bad reduction: 2, 3, 5, 11, 17, 31, 37, 41, 53, 59, 107, 109, 149, 179, 241, 419, 1789, 1879, 1405127849
regulator: 5793943444.10745291605994667597061995254227260041038396
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:29:46 UTC
last updated: 2026-09-26 20:29:46 UTC

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

(422303415191139388280, 26424942153581441228336420182026)
(966730155477531374903, 128326003979081279509471317)
(24168822175414805367281/25, -73173840472162383415832948019/125)
(323751282648029395262594150542696005437/3453859772124304, 5824362814581067129234143517452918623414264685585464213343/202981777663128345894592)
(3097294272082479180613363501629293/3203827925776, -814187311942100790828035073652013706673966959/5734608496216681024)
(5073919521485627528927029028575519/5248264046281, -26284477715473528121284575947163579312760349095/12023295338001559429)
(2224773435614129883042732887/2316484, 1200469823681049505302119999944437397857/3525688648)
(170833481105072405391835981446659/176718503641, 159279728765219403813905249363839945163204745/74288747842099939)

Commentary

2-isogenous image of D-z6old678-062, rational kernel x=-483365240359059394959. 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 20:29:47 UTC · history