Ceresa cycles of bielliptic Picard curves

We show that the Ceresa cycle $\kappa (C_t)$ of the genus $3$ curve $C_t\text{\hspace{0.17em}⁣}:y^3=x^4+2t{x}^2+1$ is torsion if and only if $Q_t=(\sqrt[3]{t^2-1},t)$ is a torsion point on the elliptic curve $y^2=x^3+1$. This shows that there are infinitely many smooth plane quartic curves over $\mathbb{C}$ (resp. $\mathbb{Q}$) with torsion (resp. infinite order) Ceresa cycle. Over $\bar{\mathbb{Q}}$, we show that the Beilinson–Bloch height of $\kappa (C_t)$ is proportional to the Neron–Tate height of $Q_t$. Thus, the height of $\kappa (C_t)$ is nondegenerate and satisfies a Northcott property. To prove all this, we show that the Chow motive that controls $\kappa (C_t)$ is isomorphic to ${\mathfrak{h}}^1$ of an appropriate elliptic curve.