Narrow class number of a the maximal totally real number field inside a cyclotomic field

Question

I am wondering how much it is known about the narrow class number of the number field $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ ($p$ an odd prime). More precisely, I am interested to know when it is odd.

By computing in MAGMA (assuming GRH) for p=7,11,$\cdots$ 83 I found that it is always $1$ apart from when $p=29$ (when it is 8).

Any help\literature to read would be much appreciated!