For a given integer $N \geq 1$, the elliptic curves endowed with a point of order $N$ are parametrized by the modular curve $Y_1(N)$. For $N \geq 4$, there is a universal elliptic curve $E \to Y_1(N)$ and you are asking whether $E$ has a nice description (e.g. Weierstrass equation). To get started with the theory, you could look at these notes by Parson, which includes the case $p=5$. If you're after explicit equations for small values of $N$, there is an article by Baaziz, Equations for the modular curve $X_1(N)$ and models of elliptic curves with torsion points, Mathematics of Computation 79 (2010), no. 272, 2371–2386.

For a given integer $N \geq 1$, the elliptic curves endowed with a point of order $N$ are parametrized by the modular curve $Y_1(N)$. For $N \geq 4$, there is a universal elliptic curve $E \to Y_1(N)$ and you are asking whether $E$ has a nice description (e.g. Weierstrass equation). To get started with the theory, you could look at these notes by Parson, which include the case $p=5$ (these notes are part of a 2003-04 working group organized by B. Conrad and Parson, for which you can find notes here). If you are after explicit equations for small values of $N$, there is an article by Baaziz, Equations for the modular curve $X_1(N)$ and models of elliptic curves with torsion points, Mathematics of Computation 79 (2010), no. 272, 2371–2386.