curve #3196

y² + xy = x³ − 174282373021373043991287685x + 5430808082827667012901303719680524143681

a-invariants: [1, 0, 0, -174282373021373043991287685, 5430808082827667012901303719680524143681]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 391512260772660933912887936166035719203490447158
discriminant (Δ): -12402470585516637860273394618549183945215593924901728307209792730412441255292048128
Faltings height: 14.4331
naive height: 196.5090
primes of bad reduction: 2, 3, 13, 19, 41, 47, 107, 109, 113, 191, 263, 619, 769, 2953, 5197, 36527, 135511
regulator: 4780132266967.74407786177663697260941044119545368540233956296429725
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:08 UTC
last updated: 2026-09-26 20:42:08 UTC

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

(32835154160018, 187374885582331738199)
(-6630649537874, -79340361623056087217)
(3126313902698, -70117779370290614761)
(9977581951946, 68448377567781264983)
(-1696403676689974/121, -95224889527438684146827/1331)
(2131642264914346, 98415447602984050964671)
(-35295347848676306688662760338110/2314867957519194169, -237401582484645230881086020183760490427279446037/3521999866552054280863785197)
(682617871641452/3721, -16677840646644579779734111/226981)
(1348115833078618/289, -337510155738676543135993/4913)
(159968948136207806/25, 63981233877457907054155831/125)
(1427381586367812674802764/6193847401, -1702915211302481680759266081072349229/487461984306101)
(35827169063794981896010286/719648519041, -211607133187395642179909400493937379569/610492951321380161)
(15493521944474, -80310456658940643409)
(5673527892304967348272250/103697524441, -13343456419578521178719996638596487549/33392780518015261)

Commentary

2-isogenous image of D-z2old141516171819-062, rational kernel x=10425279415922. Source: https://dujella.github.io/tors/z2old141516171819.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:42:08 UTC · history