Index form surfaces and construction of elliptic curves over large finite fields

Index form equations play an important role in algebraic number theory especially in computing all elements with given discriminant. Using geometric ideas Gaal, Petho and Pohst [4] gave a practical method for the solution of index form equations over quartic number fields. Continuing their investigations we introduce the notion of index form surfaces associated to quartic polynomials and show that they are either empty or elliptic surfaces.