curve #3684

y² + xy = x³ − 27227959822030883481660949639674x + 54676988991001966689322545553705118390298499044

a-invariants: [1, 0, 0, -27227959822030883481660949639674, 54676988991001966689322545553705118390298499044]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 50602109704855750262657748221866866
discriminant (Δ): 393638554341388480856201081603859227851495402245666016601959648504636572362676511999141085184
Faltings height: 16.7722
naive height: 228.7590
primes of bad reduction: 2, 3, 7, 13, 23, 29, 43, 59, 67, 313, 521, 569, 2609, 6379, 8563, 8831
regulator: 383326188492.4446298013039714687597560562369248519140361279
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:32:40 UTC
last updated: 2026-09-29 19:32:40 UTC

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

(-5776925883678636, 138485680604688976341942)
(-6124932541025889/4, 2436772945638885831649977/8)
(3447519905706132, 42227787609941215094454)
(324391752127437780/529, 2378358595273334431475868666/12167)
(1914033903030140448/361, 1671146208230278509236958834/6859)
(47630973783088585524/15625, 3527103343742079513168493518/1953125)
(108634665333670341204/36481, 11166811115984622165701488074/6967871)
(2331989675564255450580/877969, 27223925396339184547938035821446/822656953)
(828899126583462980667732/162384049, 456633535074678115030754188319657562/2069259936407)
(1406962787432949478072308/419881081, 280679220988339525465586039679882306/8603783230771)

Commentary

Specialization at t = 797/1707 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), 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 19:32:40 UTC · history