MORPHISMS AND ALGEBRAIC POINTS ON THE QUOTIENT OF THE FERMAT QUINTIC \boldmath $\mathcal{C}_{3,1}: y^{5}=x^{3}(x-1) $

MORPHISMS AND ALGEBRAIC POINTS ON THE QUOTIENT

Abstract

The aim of this article is to determine the morphisms on the quotients of Fermat quintic of affine equation $\mathcal{C}_{r,s}(5): y^5 = x^r(y-1)^s$ where $r$ and $s$ are integers such that $1 < r,s, r+s < 5$ and the algebraic points of degree $4$ on the quotient $\mathcal{C}_{3,1}: y^5 = x^3(x-1)$. We use the result of Sall in \cite{OSall} and birational morphism between the curves $ \mathcal{C}_{1,1}(5)$ and $\mathcal{C}_{3,1}(5)$ to give a parametrization of the set of quartic points the curve $\mathcal{C}_{3,1}(5)$.
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