# example of elliptic curve which does not have potential good reduction

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.

• Have you heard of LMFDB? Any elliptic curve with squarefree conductor will give you an example: if a curve has bad potentially good reduction at $p$, then $p^2$ divides the conductor. So here is a list with many examples! Oct 4, 2021 at 19:41
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, $$a=-3,b=2+p$$.