On the class numbers of certain number fields obtained from points on elliptic curves III

Abstract

We study the ramifications in the extensions of number fields arising from an isogeny of elliptic curves. In particular, we start with an elliptic curve with a rational torsion point, and show that the extension is unramified if and ``only if'' the point which generates the extension is reduced into a nonsingular point (we need to assume certain conditions in order to prove the ``only if'' part). We also study a characterization of quadratic number fields with class numbers divisible by 5.