curve #3677

y² + xy = x³ + x² − 2865810038344413995917296x + 1964429644286988908697797224491158820

a-invariants: [1, 1, 0, -2865810038344413995917296, 1964429644286988908697797224491158820]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 3682590235087728259825836306
discriminant (Δ): -160743912861066581276481712926186402389403182140197454927568199353525416596
Faltings height: 12.9932
naive height: 180.6597
primes of bad reduction: 2, 3, 7, 11, 13, 17, 37, 53, 71, 131, 199, 311, 769, 1721, 3449
regulator: 2750799115.9189545545022929235941300794876514352936999277
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:31:29 UTC
last updated: 2026-09-29 19:31:29 UTC

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

(-383025968802, 1733757958572405504)
(1037950173966, 328766027196977472)
(4731702660639/4, 3833930584977621531/8)
(-695429277029154607/2070721, 5064730877937872594995289171/2979767519)
(133684452108949/121, 509977057101565617698/1331)
(14480195553534611/18769, 1185804063999412349924429/2571353)
(72326176007424611/12769, 18671852361674659717383461/1442897)
(1476545960608361469/908209, 1095643315373821366985227509/865523177)
(14180442584727801489/2647129, 51021030284237192670183634926/4306878883)

Commentary

Specialization at t = -413/1592 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:31:29 UTC · history