Is there a database that has all the known particular values of the $j$-invariant?



Is there a database about the particular values of $j$-invariant?
What do you mean by "known"? For any $\tau\in\mathbb C$ with $\text{Im}(\tau)>0$, one can compute $j(\tau)$ to as much precision as one's computer allows, but presumably that's not what you mean. In general, if $\tau$ is algebraic and $[\mathbb Q(\tau):\mathbb Q]\ge3$, then $j(\tau)$ is transcendental over $\mathbb Q$, so you need to explain what would constitute "knowing" the value. When $\tau$ is quadratic over $\mathbb Q$, the associated ellitpic curve has CM, and $j(\tau)$ generates the Hilbert class field of $\mathbb Q(\tau)$. In that case, one can in principal determine the field and then write $j(\tau)$ in terms of a basis for that field. Is that what you mean? If so, I;m sure that many examples have been worked out over the years, but I'm not aware offhand of a place where they've been compiled. Although presumably they've been done for all imaginary quadratic fields of small class number. There's a sample computation for $\tau=\frac{1+\sqrt{-15}}{2}$ in my *Advanced Topics in the Arithmetic of Elliptic Curves* book (Example II.6.2.2), where it's shown that
$$ j\left(\frac{1+\sqrt{-15}}{2}\right) = -52515-85995\frac{1+\sqrt{5}}{2}. $$
(The field $\mathbb Q(\sqrt{-15})$ has class number 2, and its Hilbert class field is $\mathbb Q(\sqrt{-15},\sqrt5)$.)


