Every elliptic quartic curve is the complete intersection of two quadrics surfaces. Indeed, from the cohomology of the exact sequence:
0 -> IC(2) -> OP(2) -> OC(2) -> 0
we get h0(IC(2)) >= 2 and the result follows from Bezout's theorem.
Since I(C) is generated by its elements of degree two, C lies on a smooth quadric surface: an elliptic quartic is a divisor of bidegree (2,2) on a smooth quadric surface.
There are also quadric cones containing C (Ex.3).
The minimal free resolution is given by the minimal free resolution of a curve on a quadric. (it is aslo the resolution of a complete intersection). An elliptic quartic, C, has no trisecant line, so projecting from a point of C yields an isomorphism with a smooth plane cubic (this can be used to make explicit the group structure on C).