Proving $x^2 - x = y^5 - y$ is a hyperelliptic curve

Question

Greetings to one an all!

How can we prove the curve "$x^2 -x = y^5-y$" is a hyperelliptic curve?

Is a hyperelliptic curve the same as a hyperbolic elliptic curve or are there any differences?

Answer 1

A hyperelliptic curve is a curve of the form $$ y^2 = f(x) $$ where $f(x)$ is a polynomial of degree at least $5$ with $5$ distinct roots. This has nothing to do with a "hyperbolic elliptic curve," although I must admit that I do not know of anything by the name "hyperbolic elliptic curve."

If you complete the square and rename $x - \frac{1}{2}$ by $x$, your curve looks like $$ x^2 = y^5 - y + \frac{1}{4}.$$ It is easy to check that the right hand side has $5$ distinct roots, although only $3$ real roots. So this is a hyperelliptic curve.