Klein's Polyhedra for the Fifth Extremal Cubic Form

Davenport and Swinnerton-Dyer had found the first 19 extremal thernar cubic forms g_i , the meaning of which is the same as the meaning of the Markov forms for binary quadratic case. The Klein's polyhedra for the forms g₁-g₄ were recently computed by Bruno and Parusnikov. For the multiple vectors of these forms, they have computed convergents of continued fractions for some matrix generalizations of the continued fractions algorithm as well. In this paper the analogous problems for the form g₅ are studied. Namely, the Klein polyhedra for the form g₅ are computed. Their periods and fundamental domains are found. The matrix algorithm's expansions of the multiple vector of this form are computed as well.