The **minimal discriminant** (or minimal discriminant ideal) of an elliptic curve \(E\) over a number field \(K\) is the ideal \(\mathfrak{D}_{min}\) of the ring of integers of \(K\) given by
\[
\mathfrak{D}_{min} = \prod_{\mathfrak{p}}\mathfrak{p}^{e_{\mathfrak{p}}},
\]
where the product is over all primes $\mathfrak{p}$ of $K$, and \(\mathfrak{p}^{e_{\mathfrak{p}}}\) is the local minimal discriminant of \(E\) at \(\mathfrak{p}\).

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