This work studies rational \(D(q)-m\)-tuples, where \(m =\) 3, 4 or 5. First, we parametrize rational \(D(q)\)-quadruples with product of elements equal to \(m\) using the elliptic curve \(E_m\) defined via \(q,m\). For each rational nonzero \(q\) we parametrize \(m\) such that there exists a rational \(D(q)\)-quadruple with product of elements equal to \(m\). Then we construct new families of rational \(D(q)\)-quintuples. With the assumption of the Parity conjecture we prove that for each squarefree integer \(q\) in one of 99:5% of classes of residues mod 394680 there exists infinitely many rational \(D(q)\)-quintuples. Finally, we parametrize all rational \(D(q)\)-triples for rational \(q\) which are not squares, and use this parametrization to solve problems. The outline of the thesis is as follows. In the Introduction we define Diophantine \(m\)-tuples and give a historical overview of research in the field. Chapter 1 ("Some basic results on elliptic curves") gives definitions and results from algebraic geometry and elliptic curves, which are used throughout the thesis. In Chapter 2 ("Rational \(D(q)\)-quadruples") we describe rational \(D(q)\)-quadruples with product of elements equal to \(m\): We parametrize all \(m\) depending on \(q\) for which there exists such a quadruple. For such pairs \((q,m)\) we parametrize all such quadruples using triples of points on the elliptic curve \(E_m\) whose equation depends on \(q\) and \(m\). In Chapter 3 ("Rational \(D(q)\)-quintuples") we construct families of rational \(D(q)\)-quintuples for squarefree integers \(q\) in specific classes of residues. With the assumption of the Parity conjecture we prove there are infinitely many \(D(q)\)-quintuples for squarefree integers \(q\) in 99:5% of classes of residues mod 394680: In Chapter 4 ("Rational \(D(q)\)-triples") we parametrize rational \(D(q)\)-triples for each rational \(q\) which is not a square. We use this parametrization to solve problems. In Chapter 5 ("Magma codes") we give explicit codes in the programming language Magma [4] used in the thesis.