Quartic points on C_a : y^7=x^a(x-1)^a

Abstract

The goal of this work is to give a parametrization of the set of quartic points on the family
of quotients of the Fermat curve of degree $7$ of affine equation
$$ \mathcal{C}_{a} : y^{7}=x^{a}\left(x-1 \right)^{a} $$
where $a \in \lbrace 1, 2, 3\rbrace $. We use the Mordell-Weil group, the Riemann-Roch spaces
and birational morphisms to give this parametrization on $\mathcal{C}_{a}$.