The two rational roots correspond to isomorphism classes of curves with rational coefficients which are 5-isogenous to the above curve, but not isomorphic, having a different function field.
12.
3A . A uses the point R _ A to create an isogeny mapping \ phi _ A : E \ rightarrow E _ A and curve E _ A isogenous to E.
13.
The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve ( but not, in general, isomorphic to it ).
14.
An isogeny between algebraic groups is a surjective morphism with finite kernel; two tori are said to be " isogenous " if there exists an isogeny from the first to the second.
