Is there a characterization of elliptic curves over $\mathbb Q$ whose conductor is a square? Does this property have a geometric meaning?



Elliptic curves with square conductor
Square conductor implies that all reduction is additive. (The converse is not quite true, since at 2 and 3, the conductor can have higher valuation, up to valuation 5 for $p=2$ and up to 3 for $p=3$. So maybe a better question is: what does it mean for the conductor to be powerful?) A geometric implication is that every fiber of the Neron model is a group of order 1, 2, 3, or 4. So given any point $P$ on a minimal Weierstrass equation, the point $12P$ goes through the identity component of the Neron model on every fiber. But I don't know if this is the sort of geometric implication you had in mind. As Conder notes, the conductor is more of an arithmetic invariant, but Ogg's formula relates it to the geometry of the special fiber of the Neron model.


