4. In the Cartesian coordinate system, the hyperbola $\Gamma: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a, b>0)$, a line with an inclination angle of $\frac{\pi}{4}$ passes through one vertex of $\Gamma$ and another point $(2,3)$ on $\Gamma$, then the eccentricity of $\Gamma$ is $\qquad$ .
The vertices of the hyperbola are \( A(-a, 0), B(a, 0) \). Clearly, only \( k=\frac{3}{2+a}=1 \Rightarrow a=1 \). Substituting \( a=1 \) into \( 4-\frac{9}{b^{2}}=1 \Rightarrow b=\sqrt{3}, c=2 \Rightarrow e=\frac{c}{a}=2 \).