A crowning achievement of Number theory in the 20th century is a theorem of Wiles which states that for an elliptic curve $E$ over $\mathbb{Q}$ of conductor $N$, there is a non-constant map from the modular curve $X_0(N)$ to $E$. For some curve isogenous to $E$, the degree of this map will be minimal; this is the modular degree. The Jacquet-Langlands correspondence allows us to similarly parameterize elliptic curves by Shimura curves. In this case we have several different Shimura curve parameterizations for a given isogeny class. Further, this generalizes to elliptic curves over totally real number fields. In this talk I will discuss these degrees and I compare them with $D$-new modular degrees and $D$-new congruence primes. This data indicates that there is a strong relationship between Shimura degrees and new modular degrees and congruence primes.