# How to compute the ray class field of $\mathbb{Q}(i)$?

I want to verify Thm 5.6 in Silvermans Advanced Topics in the Arithmetic of Elliptic Curves that says $K(j(E),h(E[\mathbb{c}])$ is the ray class filed of $K$ modulo $\mathbb{c}$. I choose $K= \mathbb{Q}(i)$ and $\mathbb{c} = 2,3,4$. The $j$-invariant is in $\mathbb{Q}$ because its class number is $1$. And the Weberfunction is $h(x,y)= x^2$. I computed the torsion points and then we have $K(h(E[2])) = K$, $K(h(E[3])) = K(\sqrt{3})$ and $K(h(E[4])) = K(\sqrt{2})$.

Now I want to compute the ray class fields via an other method. I know something about PARI, but couldn't find there something. I am looking foreward for any ideas or help with some copmuter algebra (magma, sage, PARI). Thanks in advance!

• You can play in magma with P<x>:=PolynomialRing(Rationals()); K<i>:=NumberField(x^2+1); O:=MaximalOrder(K); c := 3*O; R:=RayClassField(c); R – reuns Sep 6 '17 at 22:34
• Thanks a lot. I tried it, but I am such a beginner that i cannot interpret the result. What does this "FldAb, defined by (<[2, 0]>, [])" mean? How is this the sturcture of $K$? – anama Sep 7 '17 at 16:07
• Anama - You can use L := NumberField(R) to get an explicit description of field over $K$. There is a similar question with good answers here: math.stackexchange.com/questions/876353/… – John M Sep 10 '17 at 13:59
• Possible duplicate of explicit example of computing ray class field for imaginary quadratic? – John M Sep 10 '17 at 14:00
• @JohnM That questions asks for an explicit example, whereas this question asks for help with the computation of a particular example. They appear to be related, but this is not (to my mind) a duplicate. I am voting to leave this question open. – Xander Henderson Sep 10 '17 at 14:44