curve #160

y² + xy = x³ − x² − 73218567298360414151945259x + 245000581765085750173970908953765294265

a-invariants: [1, -1, 0, -73218567298360414151945259, 245000581765085750173970908953765294265]
rank (lower bound): ≥ 19
conductor (N): 3193612294999847993147255601045320568117070630221566159701261290
naive height: 190.3118
Faltings height: 13.7443
discriminant (Δ): -809533846760755463057458884959863139807213987649788585257426576417408765702500
primes of bad reduction: 2, 3, 5, 13, 19, 23, 29, 31, 43, 16657, 29873, 2175553654838803, 149259273131621162380144645571
regulator: 423708814536549837.7305076307123902558268187152316566188047402169947718710324
submitted by: cocoxhuang
last updated: 2026-07-01 22:30:42

Witness: 19 independent points

(5549704953011, 3096083152855087682)
(13056506698121, 38920366941834097517)
(212900947646, -15146680585140585313)
(11070733798556, 28129323372160311527)
(-4794571897764, -22041663447857995993)
(4218305221016, -3347128863109670413)
(5564955439436, -3143585857583770243)
(22114485495959/4, -24250416298451057129/8)
(5281352058011, 2370358361443582307)
(6596721361454, -7004705656918603993)
(9217322631611, -18794053512472866493)
(5394735087236, 2647577719102945757)
(5211535300685, -2228287716765803110)
(1852282363436, 10757982573182078807)
(5751888420311, -3761956701013851193)
(5242366838660, 2287980724263201245)
(-7214909794804, 19942279400410835837)
(5258984909246, -2322195160345708513)
(8465892054285, -15228327292608313255)

Commentary

New curve of rank ≥ 19 and and log conductor 146.224, found as the specialization t = 10806/5 of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique définie sur Q de rang ≥ 22' (Acta Arith. 82 (1997)) — here at family parameters (u,v) = (4,6), sextuple {−1146, −2304, −654, 3054, 2880, −1830} — located by a discriminant-gated exhaustive specialization scan crossed with a stagedNagao-sum sieve; 19 independent points certified by positive-definite Néron–Tate height pairing. Conductor N (log N ≈ 146.22) is squarefree apart from 3², with bad primes 2, 3², 5, 13, 19, 23, 29, 31, 43, 16657, 29873, 2175553654838803,149259273131621162380144645571.

last edited by cocoxhuang at 2026-06-25 03:15:13 · history