curve #2980

y² + xy + y = x³ − x² − 9429983738201967165642054050860812482x + 11145778551240869810522211480214745282371584457183262081

a-invariants: [1, -1, 1, -9429983738201967165642054050860812482, 11145778551240869810522211480214745282371584457183262081]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 2900021013294811715348347320974526390
discriminant (Δ): 1018051610086532888073503415119531141046455847234621491636511730278642712888971416017105623446805180155208000
Faltings height: 19.8789
naive height: 267.0245
primes of bad reduction: 2, 3, 5, 7, 23, 29, 37, 47, 61, 71, 229, 349, 823, 6301, 7669, 11959, 24107
regulator: 10182493315.8246430606195410057615464105171079746283083
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:31:39 UTC
last updated: 2026-09-26 20:31:39 UTC

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

(16102815220062302425/9, -975196123774865677303843651/27)
(321734126322310023502198591/180579844, 42046627596989307700080520876998389437/2426631943672)
(2469707791210308675, -1708903060184370118904367353)
(3599702496048119496031/2025, -305760556778328950493212007589/91125)
(2473318563385986605719/121, 121689405914299609204342401177101/1331)
(72498656982705259631427120665509/24479369161569, 367256930019968118239232713990527693644275415833/121115669064035963247)
(36131051051110961884341867353509/20325914014329, 275023277182368673515310850109522285744944677/91637899541879250483)
(23613704940358326439881283719751/14607974473444, 19838165499488233061338381231175040348330793673/55832233540532958872)

Commentary

2-isogenous image of D-z6old678-020, rational kernel x=-884241588715000959. 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 20:31:40 UTC · history