I am looking for a reference for the following. I believe the set of elliptic curves $E/\mathbb{Q}$ admitting a rational 5-isogeny can be parametrized as


$$E: y^2 = x^3 + f(t)x + g(t), t \in \mathbb{Q}$$


for some polynomials $f,g \in \mathbb{Z}[t]$. The analogous parametrization for elliptic curves admitting a 7-isogeny is given by


$$\displaystyle y^2 = x^3 + u(t) x + v(t),$$


with


$$u(t) = -3(t^2 + t + 7)(t^2 - 231t + 735)$$


and


$$v(t) = 2(t^2 + t + 7)(t^4 + 518t^3 - 11025t^2 + 6174t - 64826).$$


Does anyone know what $f,g$ correspond to 5-isogenies?



Family parametrizing elliptic curves with a rational 5-isogeny
Let $\phi:E\to E'$ be an isogeny of degree $5$, and $j, j'$ be the $j$-invariants of $E$ and $E'$, respectively. Let $r$ be the rational function $$r(z)=\frac{(z^2+10z+5)^3}{z}.$$ Then there is $\tau\in\mathbb Q$ such that $$j=r(\tau)\quad\text{and}\quad j'=r(125/\tau),$$ and the converse holds too.


This result is more than 100 years old. I don't know a modern and English reference, but you can find it in Kapitel 4 of the book *Die elliptischen Funktionen und ihre Anwendungen, zweiter Teil* by Robert Fricke.


To make the example explicit: The family of these elliptic curves (up to isomorphism over $\bar{\mathbb Q}$) is $$y^2=x^3+Ax+B,$$ where $$\begin{aligned}A&=-108(t^2 + 10t + 5)(t^2 + 22t + 125),\\
B&=-432(t^2 + 4t - 1)(t^2 + 22t + 125)^2.\end{aligned}$$ Its $5$-division polynomial has the quadratic factor $$x^2 + (12t^2 + 264t + 1500)x + (36t^4 + 1584t^3 + 25128t^2 + 169488t + 400500),$$ yielding a subgroup of order $5$ which (as a group, not the individual elements) is defined over $\mathbb Q$.


