Is there a substitution that relates every Fermat curve to an elliptic curve? 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!
 A: The following reference seems to answer the question of which Fermat curves admit a non-constant map to an elliptic curve:

Neal Koblitz, David Rohrlich, Simple factors in the Jacobian of a Fermat curve,  Canadian Journal of Mathematics 30 No. 6 (1978) pp.  1183–1205, doi:10.4153/CJM-1978-099-6 (free pdf).

As a particular case of Theorem 2 there, if $N \geq 5$ is a prime $\equiv 2 \textrm{ mod } 3$ then the simple factors of the Jacobian $J_N$ of the Fermat curve $X_N$ all have dimension $(N−1)/2$. So for example, the Fermat curves $X_5$ and $X_{11}$ do not map to any elliptic curve.
The motive of the Fermat curve $X_N$ has been extensively studied: it decomposes as a direct sum of motives associated to Hecke characters of the cyclotomic field $\mathbf{Q}(\zeta_N)$, see e.g. Otsubo's work On special values of Jacobi-sum Hecke $L$-functions. Note also that every elliptic factor of a Fermat curve must have complex multiplication, essentially because $X_N$, and thus its Jacobian $J_N$, admits an action of the rather large group $\mu_N \times \mu_N$, where $\mu_N$ is the group of $N$-th roots of unity in $\mathbf{C}$. More generally, the factors of $J_N$ are CM abelian varieties.
