  We prove two arithmetic properties of Dehn surgery points on the canonical
component of the $\mathrm{SL}_2(\mathbf{C})$-character variety of the knot
$7_4$. The first is that the residue characteristics of the ramified places of
the Dehn surgery points form an infinite set, providing evidence for a
conjecture of Chinburg, Reid, and Stover. The second is that the Dehn surgery
points have infinite order in the Mordell-Weil group of the elliptic curve
obtained by a simple birational transformation of the canonical component into
Weierstrass form.
