curve #221

y² + xy + y = x³ + x² − 1771379373053841538x + 922565642019656180361670031

a-invariants: [1, 1, 1, -1771379373053841538, 922565642019656180361670031]
rank (lower bound): ≥ 14
conductor (N): 1067108442606429289452124323915075023850
naive height: 137.7015
Faltings height: 9.3620
discriminant (Δ): -11961746928396976832774801995263862052981467695744000000
primes of bad reduction: 2, 3, 5, 11, 13, 19, 31, 43, 13921, 13030973, 225715883, 47972031521
regulator: 24502377124.0149976223379472061761656606064202568292589204243053485
submitted by: Edgar Costa
last updated: 2026-07-01 22:43:39

Witness: 14 independent points

(-993905311, 41247580093347)
(-618959415, 42212245856707)
(-135133735, 34051074286467)
(523368265, 11782775553267)
(546935705, 10832299311747)
(598100585, 8777953744707)
(609494201, 8326581312963)
(687993765, 5432840767267)
(772324505, 3893973423747)
(794554505, 4089081687747)
(827810585, 4844491588707)
(919786469, 8450622415863)
(3291849305, 175391699574147)
(9056167865, 852998874939267)

Commentary

Specialization at T = -309/2 of the Mestre/Fermigier family y^2 = r(x,T) on the sextuple (a) = (-44, -60, -6, 110, 94, -94) [r = degree-<=4 remainder in p6(x-T)p6(x+T) = g^2-r, g monic degree 6, p6(x)=prod(x-a_i)]. Located by a Mestre-Nagao sieve of this family. Fourteen independent rational points were obtained via 2-descent covers; rank >= 14 follows from the positive-definiteness of their 14x14 Neron-Tate height-pairing matrix, verified independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the rank (no exact-rank/Selmer/BSD claim).

last edited by Edgar Costa at 2026-06-25 22:06:57 · history