Viewed as a theorem about parameterising elliptic curves with elliptic modular functions.
The field of level $N$ elliptic modular functions is generated by the two modular functions $j(\tau)$ and $j(N\tau)$.
These two functions satisfy an algebraic relation known as the classical level $N$ modular curve, $\Phi_N\left(j(\tau), j(N\tau)\right) = 0$ where $\Phi_N(u,v)$ is a symmetric polynomial with integer coefficents.
The modularity theorem says that every elliptic curve $E(x,y) = 0$, with conductor $N$, can be obtained as a rational mapping $x=F(u,v)$ and $y=G(u,v)$, with integer coefficients, from the curve $\Phi_N(u,v) = 0$.
The exact definition of an elliptic curve varies, but for simplicity it can be taken to be a cubic curve with rational coefficients and at least one rational point (possibly at infinity).
Therefore any elliptic curve $E(x,y) = 0$ with conductor $N$, can be parameterised by a pair of level $N$ modular functions given by $f(\tau) = F\left(j(\tau),j(N\tau)\right)$ and $g(\tau) = G\left(j(\tau), j(N\tau)\right)$.
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