Vertices: $$ (0, \pm 8) $$; foci: $$ (0, \pm 4) $$

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1. The vertices of the ellipse are given as $$ (0, \pm 8) $$. Since these points lie on the $$ y $$-axis, the major axis of the ellipse is vertical. The distance from the center $$ (0,0) $$ to a vertex is $$ a=8 $$.

2. The foci are given as $$ (0, \pm 4) $$. These also lie on the $$ y $$-axis. The distance from the center to a focus is $$ c=4 $$.

3. For an ellipse with a vertical major axis, the relationship between $$ a $$, $$ b $$ (the semi-minor axis), and $$ c $$ is:
   $$
   c^2 = a^2 - b^2
   $$
   Substituting $$ a=8 $$ and $$ c=4 $$:
   $$
   4^2 = 8^2 - b^2 \quad \Longrightarrow \quad 16 = 64 - b^2
   $$
   Solving for $$ b^2 $$:
   $$
   b^2 = 64 - 16 = 48
   $$

4. The standard form of the ellipse with a vertical major axis centered at $$ (0,0) $$ is:
   $$
   \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
   $$
   Substituting $$ b^2=48 $$ and $$ a^2=64 $$:
   $$
   \frac{x^2}{48} + \frac{y^2}{64} = 1
   $$

Thus, the equation of the ellipse is:
$$
\frac{x^2}{48} + \frac{y^2}{64} = 1
$$
The equation of the ellipse is $$ \frac{x^2}{48} + \frac{y^2}{64} = 1 $$.

Vertices: \( (0, \pm 8) \); foci: \( (0, \pm 4) \)

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