Question on Tate's algorithm for elliptic curves/the minimal model of an elliptic curve

In Silverman's "Advanced topics in elliptic curves", To prove Tate's algorithm, lemma 9.5 in chapter 4.9 states:

Let $$R$$ be a DVR with fraction field $$K$$, $$E/K$$ an elliptic curve with Weierstrass equation (...), $$W \subset P^2_R$$ the $$R$$-scheme defined by this equation.
a) If $$v(\Delta)=1$$ then $$W$$ is regular and $$W$$ is the minimal proper regular model of the elliptic curve.

However Silverman does not prove the minimality part. Why is this true? That is, why cant $$W \to \operatorname{Spec} R$$ be factored as $$W \to W' \to \operatorname{Spec} R$$ with the generic fiber of $$W'$$ isomorphic to $$E/K$$?

• If $v(\Delta) < 12$ then the Weierstrass equation is minimal.
– user208649
Jun 28 '20 at 20:52
• yes, but the minimal weierstrass equation does not necessarily yield a minimal proper regular model. Jun 28 '20 at 21:10
• Of course not, which is why Silverman first proves that it is proper and regular...
– user208649
Jun 29 '20 at 1:20
• that is not what I mean. Being minimal as a weierstrass equation is different than being minimal as a (proper regular) model. My question is: why does a minimal weierstrass equation which is proper and regular, yield a minimal proper regular model? Jun 29 '20 at 14:14

Doesn't $$v(\Delta)=1$$ imply (among other things) that the special fiber has a single component (a nodal curve that is birational to $$\mathbb P^1$$). That means that every fibral component moves, since there is only one component, so it has self-intersection $$0$$, so it cannot be blown down.