Yukawa textures in modular symmetric vacuum of magnetized orbifold models

We study quark mass matrices derived from magnetized $T^2/\mathbb{Z}_2$ orbifold models. Yukawa at three modular fixed points, $\tau=i, e^{2\pi i/3}$ and $i\infty$ are invariant under $S$, $ST$ $T$-transformations. these invariances on twisted orbifold. find that have a kind of texture structures although ones $\tau=i\infty$ not realistic. classify textures $\tau=i$ $e^{2\pi i/3}$. Moreover we investigate the conditions such matrix constructed by becomes approximately rank one matrix, which is favorable to lead hierarchical masses between third generation others. It found realistic can be obtained around $S$-invariant vacuum $ST$-invariant vacuum. As an illustrating example, show realization ratios mixing based Fritzch Fritzch-Xing matrices.