curve #2973

y² = x³ + x² − 66918725755538869720341243431384160660x + 2417654314893787075675250094261185544719625917049755856900

a-invariants: [0, 1, 0, -66918725755538869720341243431384160660, 2417654314893787075675250094261185544719625917049755856900]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 61421217691205004364755396521722860
discriminant (Δ): -2505883763249385080534153571503574718847513366714161766848317180726078213356877137988568622539766567323861292436000000
Faltings height: 21.2064
naive height: 277.7834
primes of bad reduction: 2, 3, 5, 11, 13, 37, 41, 53, 101, 199, 233, 281, 359, 691, 751, 10259, 35401
regulator: 1384175602.25488539786959236993679688679859721838493498
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:31:12 UTC
last updated: 2026-09-26 20:31:12 UTC

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

(23120691264139928064, 115021697915831925881969955330)
(104074081885249056120, 1059586084792223318319812312490)
(164060508251642330505/16, 3392342524637473083401432450925/64)
(24726012636598507651954315755/1253230801, -4156540239840012642300465631774115583543150/44365623586201)
(-6862588720731736649099880/619369, 20673989627474542281251800210326551250/487443403)
(-9210853357251728549520/1849, 4075377183193271188217083945269270/79507)
(-75893620540089655383594684840/8307775609, 36050861757244800281469774109144787121775230/757228823433523)
(-5253817547896881059040/529, -557912846129235277170069994969170/12167)

Commentary

2-isogenous image of D-z6old678-004, rational kernel x=7537968017313745980. 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 20:31:12 UTC · history