curve #3702

y² + xy = x³ − 38093853497332133063083112992074x + 88199784154175554184399411528400670974658330899

a-invariants: [1, 0, 0, -38093853497332133063083112992074, 88199784154175554184399411528400670974658330899]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 1319483920430920599678754550348754589797
discriminant (Δ): 177277782467817619005821047502012637191550394064348783530030994123011715920672477534313338060209
Faltings height: 17.0543
naive height: 229.7664
primes of bad reduction: 3, 19, 61, 71, 97, 167, 229, 269, 443, 449, 773, 1229, 1823, 1997, 2731, 2851
regulator: 38976531392307.1687673035218494587482090127764965226045692502
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:46:09 UTC
last updated: 2026-09-30 01:46:09 UTC

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

(-5310464569348647, 375146976376325132054286)
(10296725098306694091, 33040651263419675043419030637)
(-27802195024064466/25, 44928438178283668606176813/125)
(785925785633371521/256, 55843213147838387322557847/4096)
(27066979206514374515241/2483776, 3849726960380461353919913379920937/3914430976)
(73124567577145251930435/5026564, 18216246652039118342066966235034719/11269556488)
(154307259892345172097993/9406489, 56810793830789906878562425867560162/28849701763)
(503771472284577295518621/39980329, 320642625056309607030537409278390678/252795620267)
(1109318675257309848511245/271227961, 127417955895339245008868378026430202/4466853289709)
(358160944219148399763989517/610729746064, 122672606630805006769914089309217570706953/477280410711047488)
(586175666644057801225865217/411383697664, 50625906464648635936263285138555377649513/263858212612108288)

Commentary

Specialization at t = 46235/147474 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 2026-09-30 01:46:09 UTC · history