An Explicit Sextic Twist of a Supersingular Elliptic Curve in Characteristic Two

Abstract

Let E:y2+y=x3+x over F2, a supersingular curve whose automorphism group is isomorphic to SL(2,3), of order twenty-four. Using an automorphism of order six, we construct the sextic twist of E along the fiber product of an Artin-Schreier and a Kummer covering, obtaining the family Z2+Z=B(t)X3+A(t)+c, in which the constant c is not removable by a constant translation of Z over F16 when A is fixed. We analyze two members. For A(t)=t3+t and B(t)=t2(t+1), the associated elliptic surface is K3, the L-polynomial has degree eight, and the Mordell-Weil rank of the generic fiber is zero over F16(t) and four over F256(t). For A(t)=t7+t and B(t)=t17, the surface is again K3, the L-polynomial has degree fourteen, and the ranks are zero and two. In both cases, the rank follows from an exact factorization of the zeta numerator of the auxiliary curve, isolating in the non-invariant part of its cohomology the factors geometrically isogenous to E, the isogeny being defined over F256 but not over F16.