Bad reduction of an elliptic curve at a prime (reviewed)

An elliptic curve \(E\) defined over a number field \(K\) is said to have bad reduction at a prime \(\mathfrak{p}\) of \(K\) if the reduction of \(E\) modulo \(\mathfrak{p}\) is singular. There are three types of bad reduction:

• split multiplicative,
• non-split multiplicative,
• additive.

A curve has bad reduction at \(\mathfrak{p}\) if and only if \(\mathfrak{p}\) divides its discriminant.