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Burnside Curve

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BurnsidesCurve

The only known classically known algebraic curve of curve genus g>1 that has an explicit parametrization (x(t),y(t)) in terms of standard special functions (Burnside 1893, Brezhnev 2001). This equation is given by

y^2-x(x^4-1)=0.
(1)

The closed portion of the curve has area

A=(sqrt(pi)Gamma(3/8))/(4Gamma((15)/8))
(2)
=1/7sqrt((2-sqrt(2))/pi)Gamma(1/8)Gamma(3/8),
(3)

where Gamma(z) is a gamma function.

BurnsideCurveUniformized

The closed portion of this curve has a parametrization in terms of the Weierstrass elliptic function given by

x(t)=(P(1)-P(2))/(P(t)-P(2))
(4)
y(t)=4i(f(t))/([P(1/2t)-P(1)][P(1/2t)-P(2t+1)]P^'(1/2)P(t)),
(5)

where

f(t)=[P(t)-P(2t)][P(1/2t)-P(t)][P(1/2t)-P(t+2)]×[P(1/2)-P(2t+1)][P(1/2)-P(1)],
(6)

the half-periods are given by (omega,omega^')=(2,2t) and t ranges over complex values (Brezhnev 2001).

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References

Brezhnev, Y. V. "Uniformisation: On the Burnside Curve y^2=x^5-x." 9 Dec 2001. http://arxiv.org/abs/math.CA/0111150.Burnside, W. S. "Note on the Equation y^2=x(x^4-1)." Proc. London Math. Soc. 24, 17-20, 1893.

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Burnside Curve

Cite this as:

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

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