curve #3686

y² = x³ − 1518426308912492890837664761962x + 720175870235595998160629433644452843466866709

a-invariants: [0, 0, 0, -1518426308912492890837664761962, 720175870235595998160629433644452843466866709]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 172186176309954886778696599644848520
discriminant (Δ): 131444724855103147535741008773370636476180862699402521916988229065221609234113191490000
Faltings height: 15.8816
naive height: 220.0993
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 29, 31, 41, 43, 89, 107, 269, 433, 443, 449, 4651, 6367
regulator: 428166317.2371747764046064406920563868413479815865966924092
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:33:00 UTC
last updated: 2026-09-29 19:33:00 UTC

Witness: 10 independent points log in to add more points →

(-825110947090247, 37567324159599876037110)
(-321734683300562, 34284143407089487452255)
(251272242256153, 18828230090132452184910)
(710920229087173, 18915847531589385000)
(721473473611138, 464548931739172394595)
(936517883952658, 10932988333851045777645)
(-277273875475751207/256, 135489746744220032632980765/4096)
(6422721546455002/9, 2716260595248925494055/27)
(14462044121030497/4, 1649086239767378700884895/8)
(659801767248118585/961, 34009992844212199882633722/29791)

Commentary

Specialization at t = 886/615 of the Elkies-Klagsbrun Z/4 fibration E2: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = -8(80t+9), c = 2(t-2)(2t-1)(18t-1)(2t-81) (arXiv:2003.00077, eq. (6)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:33:00 UTC · history