 Trigonal linear system on a K3 surface

For a basepoint-free nonhyperelliptic nef big divisor $D$, the trigonal condition means that its general smooth curve is a trigonal curve. An elliptic pencil on a K3 surface with $D\cdot E=3$ restricts to its degree-three pencil. On a smooth quartic containing a line $\Gamma$, the hyperplane class $H$ has this property for $E=H-\Gamma$.