I am reading this paper on "Averages of Elliptic curve constants" here and in section 2.2 page no. 693 the formula for the conjectural constant in the asymptotics of the Lang-Trotter conjecture $C_{E,r}$ is defined based on the primes $l$ that either ramifies, splits or is inert in an order $\mathcal{O}$ of an imaginary quadratic number field $K=\mathbb{Q}(\sqrt{-d})$ from which the elliptic curve has complex multiplication.
But I am unfamiliar with the notion of primes ramifying, splitting and being inert in a non-maximal order. Since the character $\chi\mathcal{O}(l)$ is defined in a way that seems as if it has something to do with the Legendre symbol, is the following a correct description of those terms in this context:
i) A prime $p$ ramifies in the order $\mathcal{O}$ if and only if it divides the discriminant $D$ of $\mathcal{O}$?
ii) A prime $p$ splits in the order $\mathcal{O}$ if $\left(\frac{-d}{p}\right)=1$ and, a prime $p$ is inert if $\left(\frac{-d}{p}\right)=-1$? In this case, do all such primes actually splits or are inert, respectively in the order $\mathcal{O}$?
The author further mentions on the same page that "Since $E$ is defined over $\mathbb{Q}$, the class number of $\mathcal{O}$ must be one, and so “splitting of $l$ in $\mathcal{O}$" makes sense. So, does it mean that we can actually define these terms similarly as in the case of quadratic fields?
If somebody could provide a clarification on this matter it would be really helpful.
