curve #3697

y² = x³ − 29308830341518497796101972300x + 1931040723723252607656336563100805174562000

a-invariants: [0, 0, 0, -29308830341518497796101972300, 1931040723723252607656336563100805174562000]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 1213220854870659463310025004843200
discriminant (Δ): 403697074028694232968736559056369542215527690792931635499597961550890829831680000000
Faltings height: 15.0575
naive height: 208.2567
primes of bad reduction: 2, 3, 5, 7, 13, 19, 29, 37, 41, 79, 83, 127, 131, 163, 191, 433, 853, 883
regulator: 145460322436.3369699931235563325387062184144280510292995872
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:45:15 UTC
last updated: 2026-09-30 01:45:15 UTC

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

(94001313916130, 81184925024236407400)
(12026986356420865/1024, 41298414994636945520219775/32768)
(63842075187351490/529, 4738228713902535098045400/12167)
(160712046488469865/1764, 9669888554882894693456725/74088)
(2232410059367876944/2809, 3263754317013718086717886572/148877)
(14600119728273147721/77284, 38049238726836276872633330331/21484952)
(2143692406464508718365/25030009, 27747604262628792667853528325525/125225135027)
(4826585959622122913410/15673681, 291758738321991425286308181859800/62052103079)
(646341162093352966642210/5611657921, 121297930876731740812550643351778200/420374906520031)
(1108590519573517338899410/11107262881, 7011186557986772879363727250390200/1170605542291471)

Commentary

Specialization at t = -15653/10496 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 2026-09-30 01:45:15 UTC · history