The classical problem of the periodicity of continued fractions for elements of hyperelliptic fields has a long and deep history. This problem has up to now been far from completely solved. A surprising result was obtained in [1] for quadratic extensions defined by cubic polynomials with coefficients in the field of rational numbers: except for trivial cases there are only three (up to equivalence) cubic polynomials over whose square root has a periodic continued fraction expansion in the field of formal power series. In view of the results in [1], we completely solve the classification problem for polynomials with periodic continued fraction expansion of in elliptic fields with the field of rational numbers as the field of constants.

超椭圆场元素连分数周期性的经典问题有着悠久而深厚的历史。这个问题到目前为止还远远没有完全解决。对于由具有有理数领域中系数的三次多项式定义的二次扩展，在 [1] 中获得了令人惊讶的结果：除了平凡的情况外，只有三个（直到等价的）三次多项式的平方根具有周期性的连分数扩展在形式幂级数领域。鉴于[1]中的结果，我们彻底解决了椭圆域中以有理数域为常数域的周期性连分数展开多项式的分类问题。