The **reduction type** of an elliptic curve $E$ defined over a number field \(K\) at a prime \(\mathfrak{p}\) of \(K\)
depends on the reduction $\tilde E$ of $E$ modulo $\mathfrak{p}$. Let $\F_q$ be the ring of integers of $K$ modulo $\mathfrak{p}$, a finite field of characteristic $p$.

$E$ has good reduction at $\mathfrak p$ if $\tilde E$ is non-singular over $\F_q$. The reduction type is **ordinary** if $\tilde E$ is ordinary (equivalently, $\tilde E(\overline{\F_q})$ has $p$-torsion) and **supersingular** otherwise.

On the other hand, if the reduction of \(E\) modulo \(\mathfrak{p}\) is singular, then $E$ has bad reduction. There are two types of bad reduction are as follows.

$E$ has **multiplicative reduction** at $\mathfrak p$ if $\tilde E$ has a nodal singularity. It is called **split multiplicative reduction** if the two tangents at the node are defined over $\F_q$ and **non-split multiplicative reduction** otherwise.

$E$ has **additive reduction** at $\mathfrak p$ if $\tilde E$ has a cuspidal singularity.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2022-02-04 09:09:54

**Referred to by:**

**History:**(expand/hide all)

- 2022-02-04 09:09:54 by John Cremona (Reviewed)
- 2019-09-04 18:12:20 by John Jones (Reviewed)
- 2018-06-19 18:30:05 by John Jones (Reviewed)

**Differences**(show/hide)