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.