curve #3679

y² + xy + y = x³ − x² + 219792557452873548954739x − 20768288942605579605232414571054923

a-invariants: [1, -1, 1, 219792557452873548954739, -20768288942605579605232414571054923]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 4145713793217534004171727022
discriminant (Δ): -865877123675513667546144864225764936876515883437330647051983056520122368
Faltings height: 12.4918
naive height: 172.8545
primes of bad reduction: 2, 3, 7, 11, 13, 19, 37, 47, 59, 73, 107, 827, 1031, 3517, 5039
regulator: 142874122.51398487915473900357099740206272333545495702944
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:31:49 UTC
last updated: 2026-09-29 19:31:49 UTC

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

(124172461155, 91861160513326756)
(2558819437095, 4158803485209640408)
(2361155322765/16, 7807141743258902227/64)
(68419808060925/169, 805947481079413498006/2197)
(15494441189301201/1849, 1931678103277911268475908/79507)
(30817122310006843/841, 5410327282976213628505908/24389)
(122434998154223295/58081, 43841187465607730120553016/13997521)
(134060973317874255/185761, 57384965287306036381519496/80062991)
(5026830767603606991/53919649, 9139609820186385991126263272/395931982607)

Commentary

Specialization at t = -1391/1824 of the Elkies-Klagsbrun Z/4 fibration E2: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = -8(80t+9), c = 2(t-2)(2t-1)(18t-1)(2t-81) (arXiv:2003.00077, eq. (6)), 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 19:31:49 UTC · history