The period of a curve is the smallest positive degree of Galois-invariant divisor classes. The index is the smallest positive degree of rational divisors. We construct examples of genus one curves with prescribed period and index over certain number fields.
As Easy as $\mathbb Q$: Hilbert's Tenth Problem for Subrings of the Rationals and Number Fields
On the distribution of class groups of number fields
Elliptic curves and analogies between number fields and function fields
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