curve #218

y² + xy + y = x³ − 774334321896533858x + 184415786422239825540685056

a-invariants: [1, 0, 1, -774334321896533858, 184415786422239825540685056]
rank (lower bound): ≥ 15
conductor (N): 13737286227434328030396852859772916879098470
naive height: 135.1859
Faltings height: 9.2934
discriminant (Δ): 15022333163587070509998639856259499737597891168610762500
primes of bad reduction: 2, 5, 7, 19, 127, 151, 128983, 596538014056881445633451959507
regulator: 1779978689411.8084114367753707899730103398541064027289010096742171086
submitted by: Edgar Costa
last updated: 2026-07-01 22:41:39

Witness: 15 independent points

(979729130622811281/166926400, -959240687447310035515063389/2156689088000)
(10219787479/4, -976013371455203/8)
(4528339940910/3481, 7625308316765958588/205379)
(391446893015/11449, 15397599381839218531/1225043)
(54547127865/64, -6135098080565721/512)
(1269739826015/14884, -19806885383250721019/1815848)
(23849164131/25, 2214544291703631/125)
(-438537424, 20968123525679)
(7914992168785/1089, -22109430657093676121/35937)
(4642634492777710/208849, 316088285805959912636756/95443993)
(2450475884560/3249, 1004729461991195431/185193)
(110589032755840/22201, 1145551647025539063808/3307949)
(179225324275/169, 51800960300555549/2197)
(3796926254535/289, 7382233305884976321/4913)
(12459533075735/3844, 42451145483808864851/238328)

Commentary

Specialization at T = -490 of the Mestre/Fermigier family y^2 = r(x,T) on the sextuple (a) = (1608, -870, -1080, -642, 1542, -558) [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. Fifteen independent rational points were obtained from the low-height Mestre quartic cover together with 2-descent covers; rank >= 15 follows from the positive-definiteness of their 15x15 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 21:53:09 · history