 For that case, the canonical class is trivial (zero in the divisor class group) and so the theta characteristics of an elliptic curve "E" over the complex numbers are seen to be in 1-1 correspondence with the four points "P" on "E" with 2"P" = 0; this is counting of the solutions is clear from the group structure, a product of two circle groups, when "E" is treated as a complex torus.For "C" of genus 0 there is one such divisor class, namely the class of "-P", where "P" is any point on the curve.In case of higher genus "g", assuming the field over which "C" is defined does not have characteristic 2, the theta characteristics can be counted as

in number if the base field is algebraically closed.