curve #93

y² + xy = x³ − 2888333461706975177036x + 69181091953326695958744591394960

a-invariants: [1, 0, 0, -2888333461706975177036, 69181091953326695958744591394960]
rank (lower bound): ≥ 16
conductor (N): 2134765245448937756484936235371093624988033998420670
naive height: 160.1517
Faltings height: 11.3337
discriminant (Δ): -525428654379535377518565414066223922787670130763896329790210867200
primes of bad reduction: 2, 5, 11, 13, 19, 37, 113, 227, 7623349, 10859470707877171952191399994467277
regulator: 105196939230354.8297989405772618228792201639412619546898155735467441105
submitted by: David Renshaw
last updated: 2026-07-01 22:37:40

Witness: 16 independent points

(37416111048, 3673146956976196)
(56645298168, -9344961308712044)
(31696413064, -3078242980812300)
(38166459504, -3813140617681124)
(16981380160, -5003015355085620)
(16633886728, 5073377832734196)
(-4372609640, 9040303267636340)
(228360089416, -106387135973699596)
(4349900728, 7529900571118196)
(131565942824, 44345489428635380)
(141669139928, -50033195706517324)
(25410503148, 3492047785697456)
(32894041496, 3124735556097140)
(-13481531176, 10279596157927796)
(168559260328, 66117003092951796)
(24095569528, 3684394938234996)

Commentary

New curve of rank >= 16, found as the specialization T = 1387/2 of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve; submitted for its small conductor at this rank. 16 independent points certified by positive-definite Neron-Tate height pairing.

last edited by David Renshaw at 2026-06-12 02:09:39 · history