= Discriminant valuation obstruction to good reduction
{title2=$v_p(\Delta)\not\equiv0\pmod{12}\implies\text{bad reduction at }p$}
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.
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