Product of exponential r.v.s

[elodie]

I am reading the following from a textbook:

"
X1,...,Xn is iid Weibull with the following distribution function:

a/theta x^{a-1 } exp{ -x^a / theta}, x>0

X^a / theta is a pivotal quantity, and is distributed according to an
exponential rv with rate 1

Y=Product(i=1 to i=n, Xi^a / theta) also is a pivotal quantity because
the Xs are iid. Since we know the distribution of Y, we can find the
distribution of Y.

"
There is not much about the distribution of Y in the textbook. I would
like to know more about the distribution of Y. Does it have a name?

I appreciate your help.