Discriminant valuation obstruction to good reduction ($v_p(\Delta )\not\equiv 0(\mathrm{mod}12)⟹\text{bad reduction at }p$)

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.