4 In a regular $\triangle A B C$, $D$ and $E$ are the midpoints of $A B$ and $A C$ respectively. The eccentricity of the hyperbola with foci at $B$ and $C$ and passing through points $D$ and $E$ is $\qquad$.
$4 \sqrt{3}+1$. Establish a coordinate system with $BC$ as the $x$-axis and the perpendicular bisector of $BC$ as the $y$-axis. Let the equation of the hyperbola be $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, then $C(c, 0), A(0, \sqrt{3} c)$, so $E\left(\frac{c}{2}, \frac{\sqrt{3}}{2} c\right)$. Substituting into the hyperbola equation gives
$$
\frac{c^{2}}{4 a^{2}}-\frac{3 c^{2}}{4 b^{2}}=1
$$

which is
$$
\frac{c^{2}}{4 a^{2}}-\frac{3 c^{2}}{4\left(c^{2}-a^{2}\right)}=1,
$$

which is
$$
\frac{\mathrm{e}^{2}}{4}-\frac{3 \mathrm{e}^{2}}{4\left(\mathrm{e}^{2}-1\right)}=1
$$

Solving for $\mathrm{e}^{2}$ gives $\mathrm{e}^{2}=4 \pm 2 \sqrt{3}$. 
Since $\mathrm{e}>1$, we have $\mathrm{e}=\sqrt{4+2 \sqrt{3}}=\sqrt{3}+1$.