curve #3052

y² = x³ + x² − 8561289953050591101754799x + 9641751888652963457766757745164122330

a-invariants: [0, 1, 0, -8561289953050591101754799, 9641751888652963457766757745164122330]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 1605314370710886136520856
discriminant (Δ): 179567262762147047862570354209867984544986573296506572400834672648453072
Faltings height: 12.9090
naive height: 183.8415
primes of bad reduction: 2, 3, 7, 13, 29, 31, 47, 151, 181, 313, 661, 1009, 3049
regulator: 12754728.1889733735203215859888936732113911641919580549
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:35:02 UTC
last updated: 2026-09-26 20:35:02 UTC

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

(679112366659555/289, -7767440657668097623305/4913)
(1692165031114, -4451969470616028)
(205485798616954/121, -26031189376420418988/1331)
(2609718940795, -2252338595508577905)
(1068320186561566/169, -31695373582268857499934/2197)
(76941927682620246649/42042256, 87590050267843650911670375957/272601987904)
(154220487318292447/80089, -12334688929867619124981441/22665187)
(1671818001295, 39031615111212045)

Commentary

2-isogenous image of D-z2z4old678-023, rational kernel x=-3382740935541. Source: https://dujella.github.io/tors/z2z4old678.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 20:35:03 UTC · history