Definition. The reduction $Ẽ∕k$ of $E∕K$ is defined by the reduction of a minimal Weierstrass equation.

$E$ has good reduction if $Ẽ$ is non-singular (and so an elliptic curve) otherwise has bad reduction.

For an integral Weierstrass equation,

• If $v(\mathrm{\Delta })=0$ then good reduction.

• If $0<v(\mathrm{\Delta })<12$ then bad reduction.

• If $v(\mathrm{\Delta })\ge 12$ then beware that the equation might not be minimal.