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Ochoa Curve

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OchoaCurve

The Ochoa curve is the elliptic curve

3Y^2=2X^3+386X^2+256X-58195,
(1)

given in Weierstrass form as

y^2=x^3-440067x+106074110.
(2)

Let r_1<r_2<r_3 be the three real roots of the cubic on the right. Numerically, these are approximately -761.195736766468, 306.417090910794, and 454.778645855673. The area of the bounded real oval is

A=2int_(r_1)^(r_2)sqrt((x-r_1)(r_2-x)(r_3-x))dx
(3)
=pi/4(r_2-r_1)^2sqrt(r_3-r_1)_2F_1(-1/2,3/2;3;(r_2-r_1)/(r_3-r_1))
(4)
approx 2.28786040072697334×10^7
(5)

(E. Weisstein, Sep. 22, 2026), where _2F_1 is the hypergeometric function.

The complete set of 23 integer solutions (where solutions of the form (x,+/-y) are counted as a single solution) to this equation consists of (x,y)=(-761,504), (-745, 4520), (-557, 13356), (-446, 14616), (-17, 10656), (91, 8172), (227, 4228), (247, 3528), (271, 2592), (455, 200), (499, 3276), (523, 4356), (530, 4660), (599, 7576), (751, 14112), (1003, 25956), (1862, 75778), (3511, 204552), (5287, 381528), (23527, 3607272), (64507, 16382772), (100102, 31670478), and (1657891, 2134685628) (OEIS A141144 and A141145; Stroeker and de Weger 1994).

See also

Area, Elliptic Curve, Hypergeometric Function, Thue Equation

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References

Guy, R. K. "The Ochoa Curve." Crux Math. 16, 65-69, 1990.Ochoa Melida, J. "La ecuacion diofántica b_0y^3-b_1y^2+b_2y-b_3=z^2." Gaceta Math. 139-141, 1978.Sloane, N. J. A. Sequences A141144 and A141145 in "The On-Line Encyclopedia of Integer Sequences."Stroeker, R. J. and de Weger, B. M. M. "On Elliptic Diophantine Equations That Defy Thue's Method: The Case of the Ochoa Curve." Experiment. Math. 3, 209-220, 1994.

Referenced on Wolfram|Alpha

Ochoa Curve

Cite this as:

Weisstein, Eric W. "Ochoa Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OchoaCurve.html

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