Fix a pair of relatively prime integers $n>k\ge 1$, and a point $(��\,|\,��)\in\mathbb{C}\times\mathbb{H}$, where $\mathbb{H}$ denotes the upper-half complex plane, and let ${{a\;\,b}\choose{c\,\;d}}\in\mathrm{SL}(2,\mathbb{Z})$. We show that Feigin and Odesskii's elliptic algebras $Q_{n,k}(��\,|\,��)$ have the property $Q_{n,k}\big(\frac��{c��+d}\,\big\vert\,\frac{a��+b}{c��+d}\big)\cong Q_{n,k}(��\,|\,��)$. As a consequence, given a pair $(E,��)$ consisting of a complex elliptic curve $E$ and a point $��\in E$, one may unambiguously define $Q_{n,k}(E,��):=Q_{n,k}(��\,|\,��)$ where $��\in\mathbb{H}$ is any point such that $\mathbb{C}/\mathbb{Z}+\mathbb{Z}��\cong E$ and $��\in\mathbb{C}$ is any point whose image in $E$ is $��$. This justifies Feigin and Odesskii's notation $Q_{n,k}(E,��)$ for their algebras.

17 pages + references; numerous minor changes, in both notation and conventions