<p>The goal of this work is to give a parametrization of the set of quartic points and quintic points on the family of quotients of the Fermat curve of affine equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \mathcal {C}_{a}: y^{7}=x^{a}\left( x-1 \right) ^{a} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">C</mi> <mi>a</mi> </msub> <mo>:</mo> <msup> <mi>y</mi> <mn>7</mn> </msup> <mo>=</mo> <msup> <mi>x</mi> <mi>a</mi> </msup> <msup> <mfenced close=")" open="("> <mi>x</mi> <mo>-</mo> <mn>1</mn> </mfenced> <mi>a</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(a \in \lbrace 1, 2, 3\rbrace \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. We use the Mordell-Weil group, the Riemann-Roch spaces and birational morphisms to give this parametrization.</p>