  We study logarithmic conformal field models that extend the (p,q) Virasoro
minimal models. For coprime positive integers $p$ and $q$, the model is defined
as the kernel of the two minimal-model screening operators. We identify the
field content, construct the W-algebra W(p,q) that is the model symmetry (the
maximal local algebra in the kernel), describe its irreducible modules, and
find their characters. We then derive the SL(2,Z) representation on the space
of torus amplitudes and study its properties. From the action of the
screenings, we also identify the quantum group that is Kazhdan--Lusztig-dual to
the logarithmic model.
