curve #3211

y² + xy = x³ − 29839290725011012608171999345220110x + 1979236691612664032853709351618739821303730471684100

a-invariants: [1, 0, 0, -29839290725011012608171999345220110, 1979236691612664032853709351618739821303730471684100]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 20373912393913499123029748197579404326128537369302233310
discriminant (Δ): 8070693093581591180822228382795977201485579450221014391575872803702014639263428233406411210777889920000
Faltings height: 18.6191
naive height: 249.7570
primes of bad reduction: 2, 3, 5, 11, 13, 23, 41, 103, 179, 191, 197, 199, 227, 311, 479, 1069, 1193, 1303, 2879, 7019, 27539
regulator: 3245875390024370.2935528698440885198622306450119168534951651897286175
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:52 UTC
last updated: 2026-09-26 20:42:52 UTC

Witness: 15 independent points log in to add more points →

(104055539767636020, 979951939413316844260290)
(-77910922428754800, -61896004229566068154716990)
(-158634451561350630, -52160883831840971076503610)
(136380331090699116, 21127309924083585418274082)
(103704076736092770, 264148683676886896418040)
(103891595223881600, 731658946815154839409790)
(78542926598100360, 10959110452891654057310550)
(22249040735911952220/361, 132699329269172752257790177110/6859)
(77962943459291520, 11258495475392498916401790)
(2892044005014469440, 4909633704758863831341920190)
(29702537616691646700/841, 759365459976651459041874766890/24389)
(2253185823247335540, 3372510043584688508852607690)
(60794880538044868620/289, 347792255535348172218586768770/4913)
(43840086636328260, 27483416717952599595420090)
(120303314488017510, 11428361550061020438054900)

Commentary

Dujella (2009). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 44). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:42:52 UTC · history