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Q-factorial Gorenstein toric Fano varieties X of dimension d with Picardnumber rho(X) correspond to simplicial reflexive d-polytopes with rho(X)+dvertices. Casagrande showed that any simplicial reflexive d-polytope has atmost 3d vertices, if d is even, respectively, 3d-1, if d is odd. Moreover, itis known that equality for d even implies uniqueness up to unimodularequivalence. In this paper we completely classify all simplicial reflexived-polytopes having 3d-1 vertices, corresponding to d-dimensional Q-factorialGorenstein toric Fano varieties with Picard number 2d-1. For d even, thereexist three such varieties, with two being singular, while for d odd (d > 1)there exist precisely two, both being nonsingular toric fiber bundles over theprojective line. This generalizes recent work of the second author.

Q-factorial Gorenstein toric Fano varieties with large Picard number