curve #3069

y² + xy = x³ − 791410724196656192877500062111x + 270985332491881387056053084997768958465063385

a-invariants: [1, 0, 0, -791410724196656192877500062111, 270985332491881387056053084997768958465063385]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 1408511896423728478584743728713260910
discriminant (Δ): 763391076219577635234188903367807650464648141523228813187565619029178541380183511859200
Faltings height: 15.8121
naive height: 218.1444
primes of bad reduction: 2, 3, 5, 7, 19, 23, 29, 31, 47, 59, 61, 131, 251, 457, 6716749848486217
regulator: 46475065.844877302812819757963399342732183775003694557732
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:35:47 UTC
last updated: 2026-09-26 20:35:47 UTC

Witness: 9 independent points log in to add more points →

(-74068218887962, -18143796961075289154739)
(511534686823160, 58470570596143399925)
(357745618823558, 5800567653253786719341)
(46172989499126, 15314763171500332546349)
(355981327890944, 5862491533929390068891)
(602281137752798, 3578629072947333881861)
(380819855348144, 4982804555495324973515)
(-703211300867866, -21903697871091460615219)
(487486341852188, 1015429697665547098151)

Commentary

Kulesz - Stahlke (2001). Source: https://dujella.github.io/tors/z4old9101112.html (curve entry 37). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:35:47 UTC · history