Ceresa cycles of bielliptic Picard curves

We prove that the Chow class κ ∞ (C t) \kappa_{\infty}(C_{t}) of the Ceresa cycle of the genus-three curve C t: y 3 = x 4 + 2 t x 2 + 1 C_{t}\colon y^{3}=x^{4}+2tx^{2}+1 is torsion if and only if Q t = (t 2 - 1 3, t) Q_{t}=(\sqrt[3]{t^{2}-1},t) is a torsion point on the elliptic curve y 2 = x 3 + 1 y^{2}=x^{3}+1. In particular, there are infinitely many plane quartic curves over with torsion Ceresa cycle. Over Q I \overline{\mathbb{Q}}, we show that the Beilinson-Bloch height of κ ∞ (C t) \kappa_{\infty}(C_{t}) is proportional to the Néron-Tate height of Q t Q_{t}. Thus the height of κ ∞ (C t) \kappa_{\infty}(C_{t}) is nondegenerate and satisfies a Northcott property. To prove all this, we show that the Chow motive that controls κ ∞ (C t) \kappa_{\infty}(C_{t}) is isomorphic to h 1 \mathfrak{h}^{1} of an appropriate elliptic curve.