Let $K$ be a real quadratic number field of discriminant $D$ with fundamental unit $\varepsilon$. Further, I want to assume that each positive factor $n$ of $D$ satisfies $n \equiv 1 \pmod 4$. (For other cases I have enough information concerning my question, hence we do not need to discuss them here.)
Now let us write $D=p_1 p_2 \cdots p_k$ as product of primes. Is the sign of $N(\varepsilon)$ well understood in terms of the primes? What I know is that
- for $k=1$ we have $N(\varepsilon)=-1$,
- for $k=2$ and $(p_1/p_2)=-1$ we have $N(\varepsilon)=-1$.
What happens if $k=2$ and $(p_1/p_2)=1$? What happens for $k>2$?
Edit: Some examples for the case $D=pq$ with $(p/q)=1$ ordered by discriminant:
- $D = 5 \cdot 29$, $N(\varepsilon)=-1$
- $D = 5 \cdot 41$, $N(\varepsilon)=+1$
- $D = 13 \cdot 17$, $N(\varepsilon)=+1$
- $D = 5 \cdot 61$, $N(\varepsilon)=+1$
- $D = 13 \cdot 29$, $N(\varepsilon)=+1$
- $D = 5 \cdot 89$, $N(\varepsilon)=-1$
- $D = 5 \cdot 101$, $N(\varepsilon)=+1$