Elliptic curve invariants (reviewed)

The invariants of an elliptic curve $E$ over a number field $K$ are its

• conductor, $\mathfrak{N}$, which is an integral ideal of $K$ whose norm is the conductor norm $N(\mathfrak{N})$
• minimal discriminant, $\mathfrak{D}$, also an integral ideal of $K$, whose norm is the minimal discriminant norm $N(\mathfrak{D})$
• j-invariant, $j$
• endomorphism ring, $\text{End}(E)$
• Sato-Tate group, $\text{ST}(E)$

Each Weierstrass model for \(E\) also has a discriminant, $\Delta$, and discriminant norm, $N(\Delta)$, which are not strictly invariants of \(E\) since different models have, in general, different discriminants.