The article considers rational maps \(\mathbb{P}^1\times\mathbb{P}^1\dashrightarrow\mathbb{P}^3\) that for some bidegree \((m,n)\) send \((s:t;u:v)\) to \[ (s^mv^n:u^mt^n:s^mt^n:s^pu^{m-p}t^qv^{n-q}), \] where \(p\) and \(q\) are positive integers such that \(p<m\) and \(q<n\). A homogeneous polynomial with exponents in the indeterminates \(m\), \(n\), \(p\) and \(q\) is provided such that the image of the above rational map is the zero set of this polynomial (Theorem 3.6). Let \(J\subset \mathbb{C}[s,u,t,v]\) denote the ideal that is generated by the components of the map. The article presents the minimal free graded resolutions of the ideal \(J\) (Theorem 4.2) and the symmetric algebra of \(J\) (Theorem 5.1). Finally, the authors present a method to compute the defining equations of the Rees algebra of \(J\) (Proposition 6.2).