Two universal examples: 11.a2 and 27.a3

$E: y^2 + y = x^3 - x^2 - 10x - 20$ (LMFDB label: 11.a2)

$E: y^2 + y = x^3 - 7$ (LMFDB label: 27.a3)

$p = 2 \bmod 3 \Rightarrow a_p = 0 \Rightarrow \operatorname{End}_{\Bbb{Q}} E_p^{\mathrm{al}}$ is a quaternion algebra
$p = 1 \bmod 3 \Rightarrow \operatorname{End}_{\Bbb{Q}} E_p^{\mathrm{al}} \simeq \Bbb{Q}(\sqrt{-3})$
$\Rightarrow \operatorname{End}_{\Bbb{Q}} E^{\mathrm{al}} = \Bbb{Q}(\sqrt{-3})$