 Bad primes of a congruent number curve with odd parameter

For odd nonzero $D$, the intrinsic bad primes of $y^2=x^3-D^{2}x$ are exactly the displayed set. If $v_p(D)=2k$, the change $x=p^{2k}X$, $y=p^{3k}Y$ removes that even valuation from the parameter and gives unit discriminant at odd $p$. If the valuation is odd, after the same removal the discriminant valuation is six. Discriminant valuations of isomorphic Weierstrass models differ by multiples of twelve, so no integral model has unit discriminant. At two the original discriminant valuation is also six. Thus primes dividing a nonminimal displayed discriminant need not be intrinsic bad primes: $E_9$ is isomorphic to $E_1$ and has good reduction at three.