Let $E/\mathbb{Q}$ be an elliptic curve, let $\Delta_E \in \mathbb{Z}$ denote the discriminant of a global minimal Weierstrass equation for $E$, and let
be the conductor of $E$, where the product ranges over all prime numbers $p$ and $f_p(E)$ is the local conductor exponent at $p$. Then, for every prime number $p$,
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\begin{align*}
p \mid N_E \quad \Longleftrightarrow \quad p \mid \Delta_E.
\end{align*}
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Moreover, if $p \geq 5$, then
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\begin{align*}
f_p(E) &=
\begin{cases}
0, & p\nmid \Delta_E,\\
1, & E \text{ has multiplicative reduction at } p,\\
2, & E \text{ has additive reduction at } p.
\end{cases}
\end{align*}
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For $p=2$ and $p=3$, additive reduction can have conductor exponent strictly greater than $2$.