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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 $E/K$?

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

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