I heard $C:y^5=-x^2+x$ is quotient of the Fermat curve $x^p+y^p=1$ over $ \Bbb{Q}$.
$(1)$What is the definition of 'quotient of a given curve' here ?
$(2)$How can I confirm $C$ is quotient of $x^p+y^p=1$?
I couldn't find definition of $(1)$, so I'm at a loss. For $(2)$, only hints or sketch is also appreciated, thank you.
Here is a reference.
You need $p = 5$ (or at least $p$ divisible by $5$). Then there is a map from $F_5\colon\ x^5 + y^5 = 1$ to $C$ given by $$(x, y)\mapsto (x^5, xy).$$
As to why it's a quotient: the Fermat curve $F_5$ has a lot of automorphisms. These are generated by $$(x, y)\mapsto (\zeta_5x, y)\\(x,y)\mapsto(x, \zeta_5y)\\(x,y)\mapsto(1/y,-x/y).$$
The third map looks a bit weird, but if you rewrite $F_5$ in projective coordinates as $X^5 +Y^5 + Z^5 = 1$, then it's just $(X:Y:Z)\mapsto(Z:X:Y)$.
The map $F_5\to C$ is the quotient of $F_5$ by the group of automorphisms $H$ generated by $(x,y)\mapsto (\zeta_5x, \zeta_5^{-1}y)$, obtained by identifying points of $F_5$ that lie in the same $H$-orbit.