How to compute with the Stark conjectures?

Question

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 suggests that the Stark conjectures give explicit generators for $E$ which can then be verified using the computer algebra system PARI/GP.

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

Answer 1

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" (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" (2000)

• Xavier-François Roblot, "Checking the Brumer-Stark conjecture using PARI/GP" (2004)

Answer 2

Also useful : bnrclassfield, which replaces rnfkummer .