curve #3198

y² + xy = x³ − 364003597619839345303475845x + 2661752594229761591647083926409872269121

a-invariants: [1, 0, 0, -364003597619839345303475845, 2661752594229761591647083926409872269121]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 391512260772660933912887936166035719203490447158
discriminant (Δ): 26029928502235270351979281451310314759863469840373458951206592925295189744484352
Faltings height: 14.0866
naive height: 195.0912
primes of bad reduction: 2, 3, 13, 19, 41, 47, 107, 109, 113, 191, 263, 619, 769, 2953, 5197, 36527, 135511
regulator: 4780132266967.74407786177663697260941044119545368540233956296429715
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:15 UTC
last updated: 2026-09-26 20:42:15 UTC

Witness: 14 independent points log in to add more points →

(8843705675426, 11588270988956470823)
(-8622686155982, -71828530978734086777)
(-384019630966, -52929013493014332745)
(4037474716010, 35467066700728688759)
(-15484349407750, -67716351166827878761)
(10407391606610, 829899092205015431)
(10208626601660, 3111039289244693309)
(-17158466998714, -62095246818053375809)
(-5262124352014, -66569335769569265977)
(6398763865059122, 511849395123434923792199)
(38581505585459/4, 55507478604745283059/8)
(206949572689130, 2964893429706509319071)
(-15809709094798/49, -18082196507376854582143/343)
(13018891934150, 11376338678435580539)

Commentary

Aguirre - Peral (2009). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 62). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:42:15 UTC · history