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We show that the Ceresa cycle $\kappa(C_t)$ of the genus $3$ curve $C_t\colon y^3 = x^4 + 2tx^2 + 1$ 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 showsthat there are infinitely many smooth plane quartic curves over $\mathbb{C}$(resp. $\mathbb{Q}$) with torsion (resp. infinite order) Ceresa cycle. Over$\overline{\mathbb{Q}}$, we show that the Beilinson--Bloch height of$\kappa(C_t)$ is proportional to the Neron--Tate height of $Q_t$. Thus, theheight of $\kappa(C_t)$ is nondegenerate and satisfies a Northcott property. Toprove all this, we show that the Chow motive that controls $\kappa(C_t)$ isisomorphic to $\mathfrak{h}^1$ of an appropriate elliptic curve.

Ceresa cycles of bielliptic Picard curves