*I asked [this question on MSE](https://math.stackexchange.com/q/3043380/496634) 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](https://math.stackexchange.com/questions/1346512/substitutions-that-transform-fermat-equations-to-elliptic-curves), 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,



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> Is there a similar substitution which can relate a Fermat curve of arbitrary degree to an elliptic curve?
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This is equivalent to asking whether there is always a nonconstant morphism from a Fermat curve to an elliptic curve.


Thank you in advance!



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



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> 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](https://doi.org/10.4153/CJM-1978-099-6) ([free pdf](https://cms.math.ca/openaccess/cjm/v30/cjm1978v30.1183-1205.pdf)).
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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.


