I would like a convenient basis for the elements of a fixed abelian extension $E$ of a real quadratic field $\mathbb{Q}(\sqrt{d})$. The accepted answer to [this MO question](https://mathoverflow.net/questions/106660/kroneckers-jugendtraum-for-real-quadratic-fields/) suggests that [the Stark conjectures](https://en.wikipedia.org/wiki/Stark_conjectures) give explicit generators for $E$ which can then be verified using the computer algebra system [PARI/GP](https://en.wikipedia.org/wiki/PARI/GP).


**Question:** Given $d$, how do I use PARI/GP to find and verify the desired generators?



How to compute with the Stark conjectures?
The main reference here is the very useful User's Guide,


* C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, "[User's Guide to PARI / GP](http://math.mit.edu/~brubaker/PARI/PARIusers.pdf)" (2003)


Particularly the sections about **bnrstark** (pp. 108), **quadhilbert** (pp. 87) and **quadray** (pp. 88).


For more information and examples you may want to look at Roblot's work, for instance


* Xavier-François Roblot, Brett A. Tangedal, "[Numerical Verification of the Brumer-Stark Conjecture](http://math.univ-lyon1.fr/homes-www/roblot/resources/rota.pdf)" (2000)
* Xavier-François Roblot, "[Checking the Brumer-Stark conjecture using PARI/GP](https://pari.math.u-bordeaux.fr/Events/IHP2004/Xavier.Roblot/ihp-brumer-stark.pdf)" (2004)


