The **reduction type** of an elliptic curve $E$ defined over $\mathbb Q$ at a prime $p$
depends on the reduction $\tilde E$ of $E$ modulo $p$. This reduction is constructed by taking a minimal Weierstrass equation for $E$ and reducing its coefficients modulo $p$ to obtain a curves over $\mathbb F_p$. The reduced curve is either smooth (non-singular) or has a unique singular point.

$E$ has **good reduction** at $p$ if $\tilde E$ is non-singular over $\mathbb F_p$. The reduction type is **ordinary** (ord) if $\tilde E$ is ordinary (equivalently, if $\tilde E(\overline{\F_p})$ has non-trivial $p$-torsion) and **supersingular** (ss) otherwise. The coefficient $a(p)$ of the L-function $L(E,s)$ is divisible by $p$ if the reduction is supersingular and not if it is ordinary.

$E$ has **bad reduction** at $p$ if $\tilde E$ is singular over $\mathbb F_p$. In this case the reduction type is further classified according to the nature of the singularity. In all cases the singularity is a double point.

$E$ has **multiplicative reduction** at $p$ if $\tilde E$ has a **nodal** singularity: the singular point is a node, with distinct tangents. It is called **split** if the two tangents are defined over $\mathbb F_p$ and **non-split** otherwise. The coefficient $a(p)$ of $L(E,s)$ is $1$ if the reduction is split and $-1$ if it is non-split.

$E$ has **additive reduction** at $p$ if $\tilde E$ has a **cuspidal** singularity: the singular point is a cusp, with only one tangent. In this case $a(p)=0$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2022-02-04 09:02:19

**Referred to by:**

- ec.q.30.a3.bottom
- ec.q.kodaira_symbol
- lmfdb/elliptic_curves/elliptic_curve.py (line 1083)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 391)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 399)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 508)
- lmfdb/elliptic_curves/web_ec.py (line 634)

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

- 2022-02-04 09:02:19 by John Cremona (Reviewed)
- 2018-12-19 16:22:36 by John Jones (Reviewed)

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