We all know that the ratio of circumference of a circle to the radius is a transcendental number, but how about ellipses?

It is well known that the circumference of an ellipse with semi-axes lengths $a$ and $b$ can be expressed by the elliptic integral:

$$C=4aE(e),\qquad E(e)=\int_0^{\frac\pi2}\sqrt{1-e^2\sin^2\theta}{\,\rm d}\theta,\qquad e=\sqrt{1-\frac{b^2}{a^2}}$$

I am wondering, when $a$ and $b$ are both positive rational numbers, is the circumference always an irrational number? Moreover, is it always trancendental? If not, what are the counterexamples?

In general, I am not sure about how to study the rationality of a number if it is defined by integrals.


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