The Picard group of motivic AC(1)

We show that the Picard group Pic ( A C ( 1 ) ) of the stable category of modules over C -motivic A C ( 1 ) is isomorphic to Z 4 . By comparison, the Picard group of classical A ( 1 ) is Z 2 ⊕ Z / 2 . One extra copy of Z arises from the motivic bigrading. The joker is a well-known exotic element of order 2 in the Picard group of classical A ( 1 ) . The C -motivic joker has infinite order.