Suppose the usual modular curve $E=X_0(N)$ over $\mathbb{Q}$ has genus 1 (e.g. $N=15$). Define the conductor of $E/\mathbb{Q}$ as the ideal/integer:

$$M=\prod_{p}p^{f(E/\mathbb{Q}_p)},$$

where

$$f(E/\mathbb{Q}_p)=\begin{cases}0 & E\text{ has good reduction mod }p\\1 & E\text{ has multiplicative reduction mod }p\\ 2 & E\text{ has additive reduction mod }p\end{cases}$$

is the "exponent of the conductor of $E/\mathbb{Q}_p$" (see for example Silverman's book Advanced Topics in the Arithmtic of Elliptic Curves, chapter IV, section 10). In the case of $p=2,3$ the exponent $f(E/\mathbb{Q}_p)$ might have extra terms depending on its wild ramification.

How do you prove that $N=M$?

If $E$ had a Weierstrass equation, the proof would be straight forward since $f(E,\mathbb{Q}_p)$ is easy to calculate. You could also use the Ogg-Saito formula if you can calculate the Neron model of $E/\mathbb{Q}_p$, but this also requires a Weierstrass equation.

I know that the function field of $E/\mathbb{C}$ is equal to $\mathbb{C}(j,j_N)$ and that an algebraic relation between $j$ and $j_N$ gives an equation for $E$, but this polynomial is extremely inconvenient for calculations.

Is there a way to calculate a simple equation for $E/\mathbb{Q}$?