Bad reduction of an elliptic curve

An elliptic curve has bad reduction at a finite place when its reduced minimal Weierstrass equation is singular, equivalently when its minimal discriminant has positive valuation.

Discriminants of isomorphic Weierstrass equations of an elliptic curve differ by a twelfth power of the scale parameter, so their valuations differ by a multiple of twelve. An integral model with good reduction has discriminant valuation zero. Thus a displayed integral model with discriminant valuation not divisible by twelve proves intrinsic bad reduction. More generally, a displayed discriminant valuation between zero and eleven is already minimal, since any smaller integral-model valuation would be nonnegative and differ by twelve. Positive valuations divisible by twelve alone do not establish bad reduction; rescaling may remove them.

For odd nonzero $D$, the intrinsic bad primes of $y^2=x^3-D^{2}x$ are exactly the displayed set. If $v_p(D)=2k$, the change $x=p^{2k}X$, $y=p^{3k}Y$ removes that even valuation from the parameter and gives unit discriminant at odd $p$. If the valuation is odd, after the same removal the discriminant valuation is six. Discriminant valuations of isomorphic Weierstrass models differ by multiples of twelve, so no integral model has unit discriminant. At two the original discriminant valuation is also six. Thus primes dividing a nonminimal displayed discriminant need not be intrinsic bad primes: $E_9$ is isomorphic to $E_1$ and has good reduction at three.