curve #86

y² + xy = x³ − 52841283229846206x − 192819150610486916434364

a-invariants: [1, 0, 0, -52841283229846206, -192819150610486916434364]
rank (lower bound): ≥ 15
conductor (N): 75863584261726271948809574125392066219010
naive height: 127.1318
Faltings height: 8.6625
discriminant (Δ): 9426722265250752028761189313525609860226811622809600
primes of bad reduction: 2, 5, 7, 19, 896405796700345079, 63632217828750709343
regulator: 791888638835.30221384169538421040548888858303566962292163142474760247
submitted by: David Renshaw
last updated: 2026-07-01 22:41:38

Witness: 15 independent points

(-86289844, 1929897590322)
(232261116, 252042240242)
(-19704724, -916915922718)
(-219385604, -916819471118)
(-226833604, -349143487278)
(-223779924, 652546002082)
(-63014844, 1699073084522)
(-49541412, 1517727177010)
(649349116, 15468856647442)
(-224413004, -603000225518)
(-4649124, -229662735118)
(-15204004, -779136381518)
(-85738692, -1925430491150)
(-121604004, -2105804614478)
(-133371844, 2117209895922)

Commentary

New curve of rank >= 15 and naive height 127.13, found as the specialization T = 1043/2 (t = 1043/4) of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique definie sur Q de rang >= 22' (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve over ~1.9M specializations; 15 independent points certified by positive-definite Neron-Tate height pairing.

last edited by David Renshaw at 2026-06-11 15:29:48 · history