$v_{\id{p}}({{\mathcal{D}}}(E))$ is divisible by a prime $\ell\ge5$. By Theorem \[thm:Mazur\], $E[\ell]$ is a finite flat group scheme. If the residual representation of $E$ at the prime $\ell$ is absolutely irreducible, the modularity assumption on $E$ gives the hypothesis of Conjecture \[conj:loweringlevel\], so there must exist a mod $\ell$ Bianchi modular form of level $1$ whose Galois representation matches the residual representation of $E$ modulo $\ell$; this contradicts Theorem \[thm:noleveloneforms\].

As a corollary, we have the following version of Szpiro’s conjecture.

Let $K$ be an imaginary quadratic field of class number $1$. Assume that Conjecture \[conj:loweringlevel\] holds. Let $E/K$ be a modular elliptic curve of prime power conductor. Then $\operatorname{{\mathcal N}}({{\mathcal{D}}}_{\min}(E)) \le \operatorname{{\mathcal N}}(\operatorname{cond}(E))^{6}$, except for the curve [[](http://www.lmfdb.org/EllipticCurve/Q/11/a/2)]{} over $\Q(\sqrt{-11})$, whose conductor has valuation $1$ while its discriminant has valuation $10$. \[coro:Szpiro\]

If $E$ has potentially good reduction, then the result is well-known, see for example [@Silverman2 p. 365, Table 4.1]. Otherwise, either $E$ is semistable or is a quadratic twist of a semistable curve. Recall that there are no curves of prime conductor dividing $2$, so we can restrict to odd conductor. It is enough to consider the semistable case, since if $E$ has prime conductor $\id{p}$ and satisfies 