Elliptic curves are special mathematical objects commonly defined by a cubic equation of the form $y^2 = x^3 + ax + b$, where $a$ and $b$ are constants. Thanks to their mathematical properties, such as the ability to perform efficient arithmetic operations and the difficulty of solving discrete logarithm problem (DLP) on them, elliptic curves became ubiquitous in modern cryptography. Today elliptic curves can be found in digital signature schemes (DSA), key exchange mechanisms (KEM), zero knowledge proofs (ZKP), multi-party computation (MPC), and more. The goal of this short post is to provide a brief overview of parameters that collectively specify an elliptic curve and give a somewhat opinionated classification of existing elliptic curves. Anatomy of elliptic curves The most defining characteristic of elliptic curves is their endomorphism ring, which is also the most abstract one. Namely, it's a set of mathematical operations that can be performed on the curve. These operations include point addition, scalar multiplication, and it gives information about the properties of the curve. Order $n$ is the maximum number of points on the curve and is sometimes called cardinality. Base field $\mathbb{F}_q$ of an elliptic curve is the field over which the curve is defined. The base field size $q$ thereby defines the number of elements of the finite field $\mathbb{F}_q$.