curve #4638

y² + xy = x³ − 21190650659841824448511490x + 37759167951101197534335803398505468100

a-invariants: [1, 0, 0, -21190650659841824448511490, 37759167951101197534335803398505468100]
c-invariants: [1017151231672407573528551521, -32623921109752960396513642747669017265681]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/3ℤ
conductor (N): 1478729832407875338015623909010
discriminant (Δ): -6932288209941877830208203074656677634025259876817222666294567600128000000000
Faltings height: 13.3884
naive height: 186.5717
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 23, 29, 37, 113, 313, 433, 384919, 19910263
regulator: 7269192462.723774923847794967989412479564020233867146390814
submitted by: Maksym Voznyy
submitted at: 2026-10-19 09:13:28 UTC
last updated: 2026-10-19 09:13:28 UTC

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

(-8292917622739738520/1666681, 9625215452755632927037050910/2151685171)
(3368052387594167946820/2036329, -195463910048947325366046123336070/2905841483)
(912300244109380/361, -4039436560334810367110/6859)
(-4518623252060, -6422667246649326290)
(1487314340902180/361, 30970288097683365740890/6859)
(1525745439369146800/654481, 536906454219172832722407410/529475129)
(942244363804542605380/117050761, 24970768788529311254657853116890/1266372183259)
(-1418409692545861820/436921, -2456344447838241710847328490/288804781)
(-296569125732543830/117649, -349841047483184978142132530/40353607)
(23214073942088966020/8048569, -18178516331190421881673179370/22833790253)

Commentary

t = 685/233 for y^2 + (7t^2+309) x y - 91 (t^2-1)(t^2-25)(t^2-121) y = x^3 from https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/main/submission/board/3-r12-z3k3.ec

last edited by Maksym Voznyy at 2026-10-19 09:13:28 UTC · history