TITLE: bad reduction for elliptic curves
QUESTION [13 upvotes]: Why do elliptic curves have bad reduction at some point if they are defined over Q, but not necessarily over arbitrary number fields?

REPLY [8 votes]: It's also worth noting that "good reduction at $\mathfrak{p}$" is a local condition, so an elliptic curve may have everywhere good reduction, despite not having a Weierstrass equation that has good reduction at all primes. This is because over fields of class number greater than 1, there always exist elliptic curves that do not have global minimal Weierstrass equations. The existence, or lack of, a global minimal Weierstrass equation is governed by a certain ideal class (see Proposition VIII.8.2 in my Arithmetic of Elliptic Curves). The fact that if the class number is greater than 1, then there always exist curves with no global minimal Weierstrass equation is in the paper: "Weierstrass equations and the minimal discriminant of an elliptic curve", Mathematika 31 (1984), no. 2, 245–251. There is also a paper by Bekyel that describes the density of curves having (or not having) global minimal Weierstrass equations: "The density of elliptic curves having a global minimal Weierstrass equation", J. Number Theory 109 (2004), no. 1, 41–58. The moral is that you can produce curves with everywhere good reduction by writing down a specific Weierstrass equation, but to determine whether a given curve has everywhere good reduction is done via local calculations, and the associated elliptic scheme having everywhere good reduction may need to be patched together using more than one Weierstrass equation.