We describe a tabulation of (conjecturally) modular elliptic curves over the field
$\mathbb{Q}\left(\sqrt{\mathfrak{5}}\right)$ up to the first
elliptic curve of rank
$\mathfrak{2}$.
Using an efficient implementation of an algorithm of Lassina
Dembélé, we computed tables of Hilbert modular forms of weight
$\left(\mathfrak{2},\mathfrak{2}\right)$ over
$\mathbb{Q}\left(\sqrt{\mathfrak{5}}\right)$,
and via a variety of methods we constructed corresponding elliptic
curves, including (again, conjecturally) all elliptic curves over
$\mathbb{Q}\left(\sqrt{\mathfrak{5}}\right)$ that
have conductor with norm less than or equal to 1831.
