Bicuspid curve

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Biscuspid curve where a = 1
Biscuspid curve where a = 1

The biscuspid is a quartic plane curve with the equation

(x^2-a^2)(x-a)^2+(y^2-a^2)^2=0 \,

where a determines the size of the curve.

The bicuspid has only the two nodes as singularities, and hence is a curve of genus one, with j-invariant −4096/11. It is therefore isomorphic to an elliptic curve. If we substitute

x = \frac{a(2u-1)}{2u^2-2u+1}

and

y = \frac{a(2v+1)}{{2u^2-2u+1}}

into the equation of the bicuspid and factor, we obtain

v^2 + v = u^3 - u^2\,

which is the equation of an elliptic curve in Tate-Weierstrass form. This curve is a very well known one, being one of the three curves of conductor 11, which is the smallest conductor for elliptic curves. This means that the bicuspid curve can be parametrized by modular forms of level 11.

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