According to Frey's work, the minimal discriminant of the Frey equation
$$y^2 = x(x - a^p)(x + b^p)$$
for a nontrivial solution $(a, b, c) \in \mathbb Z^3$ to the Fermat equation $a^p + b^p + c^p = 0$ and $p \geq 5$ prime is
$$\Delta = \frac{(abc)^{2p}}{2^8}$$
I fail to see why this discriminant is minimal. For example, we are assuming throughout all this that $b$ is even and $a$ and $c$ are odd, so what if $p = 11$? Then the valuation of the $\Delta$ at the prime divisor $2$ is at the very least $14 > 12$, and according to Silverman's book, the valuation at every prime divisor of the minimal discriminant must be less than $12$. Where am I going wrong?
It is not true that the valuation of every prime divisor of the minimal discriminant has to be less than $12$. By way of example (there are many simpler), the elliptic curve defined by $$ y^2 + xy = x^3 − 424151762667003358518x − 6292273164116612928531204122716 $$ has minimal discriminant divisible by, for example, $2^{33}$, $7^{18}$ and $13^{27}$.