curve #3169

y² + xy + y = x³ − 51012395091150795883443767656668298x + 76434974688414573638384904319711864744626666727159256

a-invariants: [1, 0, 1, -51012395091150795883443767656668298, 76434974688414573638384904319711864744626666727159256]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 39251473210755654248832430609878927570345791613367110
discriminant (Δ): -2515380058113317128720512716704927677579331502520697286335580360030806173887175045500623830591024628525687500
Faltings height: 19.4795
naive height: 257.0597
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 79, 5407, 601493999163407249771
regulator: 77581184036.409962403371345458682685510355523688170805493887026
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:37 UTC
last updated: 2026-09-26 20:40:37 UTC

Witness: 12 independent points log in to add more points →

(457917641278676257, 386128984291844301517862723)
(270901045199761935, 287221831349545531747622527)
(-397254005700385953, 184415027696538474481219633)
(331187106878779291, 309623241474837135248838809)
(272065748698447585, 287566503142636601677121387)
(-419789818786899680, 154507705600056059101944152)
(768976913240434780, 701372420641720230475880867)
(95702143817093695, 269127303497606029751993777)
(-351066191606211200, 225999296032216298947768472)
(-221402129973000095, 277265825716404359585558867)
(329196849140580055, 308734681486552150048900727)
(183025514774816392, 270609415057442326162910063)

Commentary

Eroshkin (2006). Source: https://dujella.github.io/tors/z3old891011121314.html (curve entry 15). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:37 UTC · history