  In this article we construct a minimal symplectic 4-manifold R that has small
Euler characteristic (e(R)=8) and two essential Lagrangian tori with nice
properties. These properties make R particularly suitable for constructing
interesting examples of symplectic manifolds with small Euler characteristic.
In particular, we construct an exotic symplectic CP^2# 5(-CP^2), the smallest
known minimal symplectic 4-manifold with pi_1=Z, the smallest known minimal
symplectic 4-manifolds with pi_1=Z/a + Z/b for all a,b in Z, and the smallest
known minimal symplectic 4-manifold with pi_1=Z^3. We use the pi_1=Z example to
derive a significantly better upper bound on the minimal Euler characteristic
of all symplectic 4-manifolds with a prescribed fundamental group.
