Abstract
We present a computational method for obtaining generic forms of sextics with a given configuration of local singularities. Using this method, we first complete the classification of topological types of local singularities appearing on reduced sextics. Next we give the list of possible configurations of singularities containing at least one non-simple singularity and satisfying $\rho(5)\ge 7$, and show that such a sextic is of torus type, which has been conjectured by the third author.
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