*I asked this question on MSE but didn't get any response, so I'm asking here. I apologize in advance if this question is not research level.*

A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,z\in \mathbb R$. In this question, the OP provides a substitution which relates a Fermat Curve of degree $n=3,4$ to two different elliptic curves. To transform the Fermat Curve of degree $3$, the substitutions $$ a=\frac{12z}{x+y},\quad b=\frac{36(x-y)}{x+y} $$ produce $b^2=a^3-432$, an elliptic curve. Similarly for the Fermat Curve of degree $4$, the substitutions $$ a=\frac{2(y^2+z^2)}{x^2},\quad b=\frac{4y(y^2+z^2)}{x^3} $$ give $b^2=a^3-4a$. However, the substitutions used are not at all obvious, which leads me to wonder,

Is there a similar substitution which can relate a Fermat curve of arbitrary degree to an elliptic curve?

This is equivalent to asking whether there is always a nonconstant morphism from a Fermat curve to an elliptic curve.

Thank you in advance!

On special values of Jacobi-sum Hecke $L$-functions). So your question is about rationality properties of these Hecke characters. $\endgroup$ – François Brunault Dec 20 '18 at 9:24Simple factors in the Jacobian of a Fermat curve. If I understand Theorem 2 correctly, then if $N \geq 5$ is a prime $\equiv 2 \textrm{ mod } 3$ then the simple factors of the Jacobian of $X_N$ all have dimension $(N-1)/2$. So for exemple $X_5$ and $X_{11}$ do not map to any elliptic curve. $\endgroup$ – François Brunault Dec 20 '18 at 9:38