In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four non-zero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups $\mathbf{Z}/2\mathbf{Z} \times \mathbf{Z}/k\mathbf{Z}$ for $k=2,4,6,8$, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.

Let $X$ be a normal $\mathbf{Q}$-factorial projective variety with at most log canonical singularities. We shall give a sufficient condition for the existence of at most finitely many $K_{X}$-negative extremal rays $R(\subset \overline{\mathrm{NE}}(X))$ of divisorial type. As an application, we show that for a nonisomorphic surjective endomorphism $f\colon X\to X$ of a normal projective $\mathbf{Q}$-factorial terminal 3-fold $X$ with $\kappa(X) > 0$, a suitable power $f^{k}\ (k > 0)$ of $f$ descends to a nonisomorphic surjective endomorphism $g\colon X_{\textit{min}}\to X_{\textit{min}}$ of a minimal model $X_{\textit{min}}$ of $X$.