curve #3156

y² = x³ + x² − 98157972552649846551441608681894049x + 17173523831080699917850997782683206552726896135031487

a-invariants: [0, 1, 0, -98157972552649846551441608681894049, 17173523831080699917850997782683206552726896135031487]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 725195062080830777445016962580423655128128
discriminant (Δ): -66881671631823734629798952883960073044859214239556926932282052584068372188446835412767426010633622581280768
Faltings height: 19.1992
naive height: 254.0736
primes of bad reduction: 2, 3, 11, 13, 29, 41, 47, 53, 61, 89, 193, 233, 257, 419, 509, 683, 3037, 14519, 22129
regulator: 18967193211569.4063134526550983582686959905521386448037001321
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:00 UTC
last updated: 2026-09-26 20:40:00 UTC

Witness: 11 independent points log in to add more points →

(128855060092964637, 81638432561593390605208164)
(-7938783333687338421/25, 15969799545743675692435201008/125)
(24817026840736474269, 123620432467089668380394853924)
(12309421217494150611/49, 31377430254863109626702062128/343)
(4557714551401009933251/16129, 224391997963421310525101749971792/2048383)
(1551372858474736219998651/64009, 1932138513456898568279766310524953808/16194277)
(1677453647094357397905987/261121, 2170065217864825235714208165103733424/133432831)
(107899012783006936461819/51529, 35077189914780926191611962760851760/11697083)
(-1414306374225384339701/29241, 738407823739449169949563410382256/5000211)
(-1034333775630154712091/3844, 37001289006451624962607023421785/238328)
(25439111806571211796699713/161086864, 153143561548162495891268778410148849033/2044514477888)

Commentary

Voznyy (2022). Source: https://dujella.github.io/tors/z4old9101112.html (curve entry 14). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:00 UTC · history