Consider the complete fan $\Delta$ in $\mathbb R^2$ with edge vectors $$ v_1=e_1,\qquad v_2=-a_1e_1+a_2e_2, \qquad v_3=-b_1e_2-b_2e_2\,, $$ where $a_1,a_2$ and $b_1,b_2$ are respectively relatively prime positive integers. Then the corresponding toric variety $X(\Delta)=\mathbb C^3\setminus\{0\}/\mathbb C^*$ where the $\mathbb C^*$ action on by pointwise multiplication under the identification $$\mathbb C^*\cong\left\{\left(t^{\frac{a_1}{a_2}+\frac{b_1}{b_2}},t^{\frac{1}{a_2}},t^{\frac{1}{b_2}}\right)\, \mid\, t\in\mathbb C^*\right\}$$
Is there a specific name for such a toric variety? Is this toric variety a weighted projective space of some kind?