I'm looking for an example of elliptic curve which does not have potential good reduction.
Let $E$ be an elliptic curve over local field $K$(whose integer ring is $R$), then, it is well known that
$E/K$ has potential good reduction is equivalent to it's $j$-invariant $j(E)$is in $R$.
So, for example, if I could find example of $E/\mathbb{Q}_p$, which satisfies $j(E)＝1/p$, that's it. There exists elliptic curve which has given $j$-invariant, so I know I know there exists titled elliptic curves, but I'm searching for an simplest one.
Thank you in advance.
Best Answer
Consider the elliptic curve given by $y^2=x^3-ax/p^2+b/p^3$ for $p \geq 5$, where $a,b \in \mathbb{Z}_p^{\times}$ are such that $p|4a^3+27b^2$ and $4a^3+27b^2 \neq 0$. For instance, $a=-3,b=2+p$.