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Let $C$ be a hyperelliptic curve defined over $\mathbb{Q}$, whose Weierstrasspoints are defined over extensions of $\mathbb{Q}$ of degree at most three, andat least one of them is rational. Generalizing a result of R. Soleng (in thecase of elliptic curves), we prove that any line bundle of degree $0$ on $C$which is not torsion can be specialised into ideal classes of imaginaryquadratic fields whose order can be made arbitrarily large. This gives apositive answer, for such curves, to a question by Agboola and Pappas.

From Picard groups of hyperelliptic curves to class groups of quadratic fields