Simple Complex Tori of Algebraic Dimension 0

Using Galois theory, we explicitly construct (in all complex dimensions $$g\ge 2$$ ) an infinite family of simple $$g$$ -dimensional tori $$T$$ that enjoy the following properties: $$\bullet$$ Picard number is $$0;$$ in particular, algebraic dimension $$0$$ ; if $$T^\vee$$ dual , then $$\mathrm{Hom}(T,T^\vee)=\{0\}$$ automorphism group $$\mathrm{Aut}(T)$$ isomorphic to $$\{\pm 1\} \times \mathbb Z^{g-1}$$ endomorphism algebra $$\mathrm{End}^0(T)$$ a purely imaginary field degree $$2g$$ .