I'm taking a course on elliptic curves, in which the lecturer briefly mentioned that the Tamagawa number $c_K(E)=[E(K):E_0(K)]$ satisfies $c_K(E)=\mbox{ord}(\Delta)$ or $c_K(E) \leq 4$.
He said that this is only true if we use a minimal Weierstrass equation to get your reduction (unlike previous results we'd done). Otherwise, he said we could find Weierstrass equations with arbitrarily large $[E(K):E_0(K)]$. (Of course we don't actually mean $E_0(K)$ here, we mean a thing depending on our choice of non-minimal Weierstrass equation.)
I can't think of an example of this. I can think of examples where if we use non-minimal Weierstrass equations then $c_K$ isn't well-defined: let $E/\mathbb{Q}_3$ be given by
$$E:y^2=x(x-1)(x-2)=x^3-3x^2+2x.$$
This has good reduction $\tilde{E}/\mathbb{F}_3$
$$\tilde{E}:y^2=x^3+2x=x(x-1)(x-2),$$
so $E_0(\mathbb{Q}_3)=E(\mathbb{Q}_3)$.
However, by substituting $y=\frac{1}{3}^3$, $x=\frac{1}{3}^2$, $E$ is also given by the equation
$$E:y^2=x(x-9)(x-18)$$
which reduces to $y^2=x^3$, which has $'E_0(K)`=E(\mathbb{Q}_3)\setminus\{(0,0)\}$.
So $[E(\mathbb{Q_3}):\;'E_0(K)`]=2$. I don't see how to make this situation any worse though.