# Revision history [back]

### How to compute a cyclic subgroup of a class group?

Hello, I am quite new to computing in Sage and I am stuck at this point. Please help me out. I am going to use algebraic number theory notation without explaining them in detail. All computations are in quadratic fields.

What I have :

• $\Delta_p=p^2\Delta_K$ where $p$ is a prime, $\Delta_K=-pq$ and $\Delta_K\equiv 1$ mod $4$
• I computed the class group, $C(\Delta_p)$ in Sage

What I want :

• Set $f=[(p^2,p)]$ in $C(\Delta_p)$ where $(p^2,p)$ is the standard representation of an ideal of norm $p^2$
• Set $F=\langle f \rangle$

My question is :

• How to get that cyclic subgroup $F$?

I have been looking at AbelianGroup and ClassGroup in Sage but I am not quite understanding how to use these to actually get what I want.

Any help in this matter would be greatly appreciated.

 2 retagged FrédéricC 2604 ●3 ●29 ●54

### How to compute a cyclic subgroup of a class group?

Hello, I am quite new to computing in Sage and I am stuck at this point. Please help me out. I am going to use algebraic number theory notation without explaining them in detail. All computations are in quadratic fields.

What I have :

• $\Delta_p=p^2\Delta_K$ where $p$ is a prime, $\Delta_K=-pq$ and $\Delta_K\equiv 1$ mod $4$
• I computed the class group, $C(\Delta_p)$ in Sage

What I want :

• Set $f=[(p^2,p)]$ in $C(\Delta_p)$ where $(p^2,p)$ is the standard representation of an ideal of norm $p^2$
• Set $F=\langle f \rangle$

My question is :

• How to get that cyclic subgroup $F$?

I have been looking at AbelianGroup and ClassGroup in Sage but I am not quite understanding how to use these to actually get what I want.

Any help in this matter would be greatly appreciated.