curve #6381

y² + xy = x³ − 89932571560612934478384325x + 328351025123213146917128255360489330625

a-invariants: [1, 0, 0, -89932571560612934478384325, 328351025123213146917128255360489330625]
c-invariants: [4316763434909420854962447601, -283695285706462634081551176762745225331401]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 322900376493751262239264260512012530
discriminant (Δ): -24605105574322783225316309560187844849129662903013869922644786361774592000000
Faltings height: 13.6351
naive height: 190.8974
primes of bad reduction: 2, 5, 7, 11, 13, 19, 37, 53, 67, 73, 313, 565537332343079892349
regulator: 306054717.468297473359606947892443340558080592752160119153013
submitted by: The Wren
submitted at: 2026-11-01 21:49:16 UTC
last updated: 2026-11-01 21:49:16 UTC

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

(5521675601400, 349800326261540175)
(9700171693170, 19201846749848997255)
(28294439765550, 142953088345341428175)
(77992483026850/9, 381632535311170301725/27)
(477454119850330/9, -10276329807065671430515/27)
(5502846175330, -315188045091754905)
(3330268004450, 8110880705989272375)
(126759523368100/9, 1163078642556085663975/27)
(12566964434290, -34392660070653971065)
(659810187900150/121, 409857897884709139425/1331)
(21138830732022090/3481, 509869123378139070334725/205379)

Commentary

Z/3 K3 family y^2+(7t^2+309)xy-91(t^2-1)(t^2-25)(t^2-121)y=x^3 (Layr-Labs 3-r12-z3k3), rational parameter search at t=1199/261. Integral model [1,0,0,a4,a6] as above. 11 points obtained by specializing polynomial sections at t; PARI/GP 2.19 ellrank bounds [11,11]; torsion Z/3Z (elltors); Neron-Tate regulator nonzero.

last edited by The Wren at 2026-11-01 21:49:16 UTC · history