curve #64

y² + y = x³ + 18128458663461957134862581373

a-invariants: [0, 0, 1, 0, 18128458663461957134862581373]
rank (lower bound): ≥ 16
conductor (N): 141972917837550720141347359033819641029697926227598433712323
naive height: 143.6577
Faltings height: 10.0291
discriminant (Δ): -141972917837550720141347359033819641029697926227598433712323
primes of bad reduction: 3, 41, 1768630113508483622913422573
regulator: 359787206510825092.8999100746454939974782593912647860402619044739747445
submitted by: David Renshaw
last updated: 2026-07-01 22:37:39

Witness: 16 independent points

(1061832153, 139016765771325)
(-2069581821, 96250143600728)
(2323084809, 175115687339501)
(2437726383, 180595326275964)
(3097225419, 218722516138736)
(7619958189, 678654480211793)
(13493940633, 1573274288396285)
(15245095569, 1887136964766261)
(6376192377/4, 1191407043318535/8)
(17844824169, 2387591696553138)
(5948344741/9, 3664166411156266/27)
(7271819311/9, 3687841575833417/27)
(-11236359299/9, 3434674826973187/27)
(31140274567/9, 6588845927720308/27)
(-26120241831/16, 7512202525369347/64)
(-1185892731/25, 16830196043015853/125)

Commentary

Provenance: Noam D. Elkies, "Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds", ANTS XVI. This is the minimal model (7) for E_{16D}, y^2 + y = x^3 + (D-1)/4 with D = 72513834653847828539450325493. The submitted points are the 16 independent points P_i in table (10). This is the 3-isogenous companion of the E_{-432D} model already on the board and has smaller naive height.

last edited by David Renshaw at 2026-05-29 04:47:26 · history