curve #3692

y² + xy = x³ − 33450850199768465127278561610538009889893x + 2323101587147741941087535816327735979138519566431276135710977

a-invariants: [1, 0, 0, -33450850199768465127278561610538009889893, 2323101587147741941087535816327735979138519566431276135710977]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 608480130869274673011093246882337748935299188663528462593209259950
discriminant (Δ): 64111099268509703613834758035457656267019943049972595444243190601816735444520891882320080122723974075322789048412438528000
Faltings height: 22.1738
naive height: 291.5463
primes of bad reduction: 2, 3, 5, 7, 17, 19, 23, 43, 53, 61, 101, 167, 233, 563, 587, 607, 1033, 1123, 2473, 9161, 67271, 97987, 3788861, 25217561
regulator: 46986716652564089289785.9607674468555366612846432200826597478528798057339
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:44:19 UTC
last updated: 2026-09-30 01:44:19 UTC

Witness: 17 independent points log in to add more points →

(113302624606484243803874/25, 38107587432616350211145093631763147/125)
(39322883381526566705199332/776161, 595498011341895017950141834676640745909/683797841)
(51256684414334643606665846/326041, 181427652099087592235171060194960537649/186169411)
(268791699788850814558563254/2325625, 103007940858634817811539087732713536083/3546578125)
(905350392277787644959969638/2193361, 24917808969814967440476285120622816878115/3248367641)
(1165970963818321501349934146/2886601, 36276872309727260526879383787220927741565/4904335099)
(2580394957467082543340003138/45333289, 237114221891353146691599726646380877955123/305229034837)
(13220549639032227459986993426/939361201, 39213256775040622660585501267416103776808511/28790481449449)
(25877568741356340335106048338/316021729, 2048998337466828739552571691978331008138967/5617918276433)
(155206370269741268955627155426/1286011321, 9660474459839410064052374619358501338789979/46117651982381)
(4690907690999027441475541117574/27144210025, 5836752898849931791089107848762717096510721333/4472144322668875)
(30726274853073717805635900578162/167036507401, 105632610977716320273281117079259406126878068739/68267987611296101)
(205002713491443787711409319628322/364754394601, 2795682673504162487854843725810944393403922094771/220293051864879349)
(233814792062015796011478603499682/2662311092281, 1087780105025555529591651364495681025371872035621/4343983854520124179)
(256901129273065608088952734697384/2698716271729, 167105371734145469711842505574941304132678686855/4433389020722151433)
(3726473092714096973020891979920238/32232382505449, 5991955985619761724482823448392050244042835277327/182994742443988418293)
(27734249652086970787051296744446702018/17506931879220169, 145124051086161465048465200447318453118621702508652463613/2316408035524282065885947)

Commentary

Specialization at t = 2865/12073, u = 11/5 of the Elkies-Klagsbrun family y^2 = x^3 + 2A(t,u)x^2 + B(t,u)x (arXiv:2003.00077, Sec. 9, eqs. (3)-(4)), 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 01:44:19 UTC · history