8. Given that the parabola $P$ has the center of the ellipse $E$ as its focus, $P$ passes through the two foci of $E$, and $P$ intersects $E$ at exactly three points, then the eccentricity of the ellipse is $\qquad$

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8. $\frac{2 \sqrt{5}}{5}$

Analysis: Let the ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. Without loss of generality, assume the parabola $P$ opens upwards, then the parabola $P$ must pass through the lower vertex and the two foci of the ellipse $E$. Therefore, the equation of the parabola $P$ is: $\frac{c^{2}}{b}(y+b)=x^{2}$. Since the parabola $P$ has the center of the ellipse $E$ as its focus, we have $\frac{c^{2}}{4 b}-b=0$, solving this gives $e=\frac{2 \sqrt{5}}{5}$