The length of the minor axis is equal to one-fourth of the distance between the foci. Mathematically, this can be expressed as:

$ 2b = \frac{1}{4}(2ae) $

This simplifies to:

$ b = \frac{ae}{4} $

Given that the relationship between $ b $, $ a $, and $ e $ is:

$ \frac{b^2}{a^2} = 1 - e^2 $

Substitute $ b = \frac{ae}{4} $ into the equation:

$ \left(\frac{ae}{4}\right)^2 = a^2 \cdot \left(1 - e^2\right) $

Expanding and simplifying gives:

$ \frac{a^2 e^2}{16} = a^2(1 - e^2) $

Divide both sides by $ a^2 $:

$ \frac{e^2}{16} = 1 - e^2 $

Rearrange and solve for $ e^2 $:

$ \frac{e^2}{16} + e^2 = 1 $

$ \frac{17e^2}{16} = 1 $

Solve for $ e $:

$ e^2 = \frac{16}{17} $

$ e = \frac{4}{\sqrt{17}} $

Thus, the eccentricity of the ellipse is:

$\frac{4}{\sqrt{17}}$