Let \(p\) be a prime number and \(k\) a positive integer. Let \(E_{p^k}\) and \(E_{-p^k}\) be the elliptic curves over \(\mathbb Q\) given by the Weierstrass equations \[ E_{p^k} : Y^2=X(X^2+p^k)\quad \text{and} \quad E_{-p^k} : Y^2=X(X^2-p^k). \] The author determines the sets \(E_{p^k}(\mathbb Z)\) and \(E_{-p^k}(\mathbb Z)\) of the integer points of \(E_{p^k}\) and \(E_{-p^k}\). For this aim, he is led to solve a finite number of quartic elliptic equations, using a reduction through an unramified map. Coombes and Grant have already used a similar reduction in order to compute the rational points on some families of curves of genus two. Let us mention the results obtained by the author concerning the set \(E_{-p^2}(\mathbb Z)\). Let us set \[ \begin{aligned} A&= \{ (-a^2,\pm ab) \mid (a,b)\in \mathbb Z^2 \;\text{with} \;a^4+b^2=p^2\},\\ B& =\{ (a^2,\pm ab) \mid(a,b)\in \mathbb Z^2 \;\text{with} \;b^2-a^4=-p^2\}.\end{aligned} \] If \(p\) is distinct from \(5\) and \(29\), we have \[ E_{-p^2}(\mathbb Z)=\{ (0,0), (\pm p, 0)\} \cup A\cup B. \] Suppose \(p=5\) or \(p=29\): the groups \(E_{-25}(\mathbb Q)\) and \(E_{-841}(\mathbb Q)\) are of rank one, and we have \[ \begin{aligned} E_{-25}(\mathbb Z)&= \{ (0,0),(-4,\pm 6), (\pm 5,0), (45,\pm 300)\},\\ E_{-841}(\mathbb Z)&= \{ (0,0),(\pm 29,0),(284229,\pm 151531380)\}.\end{aligned} \]