Extremal elasticity of quadratic orders

We study how large and small elasticity can be for orders belonging to a fixed quadratic field, in terms of the corresponding conductors. For example, we show that if $K$ is an imaginary quadratic field, then the order of conductor $f$ in $K$ has elasticity exceeding $(\log f{)}^{c_1\log \log \log f}$ for all $f$ that are sufficiently large. On the other hand, this elasticity is smaller than $(\log f{)}^{c_2\log \log \log f}$ for infinitely many $f$. Here $c_1,c_2$ are universal positive constants. The proofs borrow methods from analytic number theory previously employed to study statistics of the multiplicative groups $(\mathbb{Z}\mathrm{/}m\mathbb{Z}{)}^{\times }$.