The "regulator" access routine should indeed at least allow to produce the
regulator with the precision that pari used.
If you look at the source of "regulator" you can see how to get the
regulator to better precision. If K is your number field, then:
K.pari_bnf().bnf_get_reg().sage()
gets you the regulator straight from pari (pari should really be choosing
an appropriate working precision here by itself).
I'm not so sure that computing the regulator this way is efficient for
quadratic extensions: there are way better special algorithms for that
(continued fractions do a lot of the work already, for instance). The
general algorithm determines the regulator by determining the class group
and unit group. So finding the fundamental unit through a regulator
computation wouldn't really make much sense.
On Sunday, 26 December 2021 at 12:21:18 UTC-8 Michael Beeson wrote:
> I want to compute the regulator of a real quadratic field Q(sqrt d) to
> high precision,
> accurately enough to compute the fundamental unit. The default
> breaks at d = 331 where fundamental unit needs more than 53 bits (the
> precision of doubles). The documentation says that Pari computes to a
> higher precision than
> SageMath. Also somewhere it says that if you get a good enough
> approximation to the regulator, it's trivial to refine it to high accuracy.
> It refers to "the tutorial" without a link; I read the Pari-GP tutorials
> on algebraic number theory without finding any explanation of that remark.
> So actually there are two questions here: point me to an explanation of
> refining the computation of the regulator, and secondly, fix the
> following code
> so that it doesn't print "oops" when d = 331.
>
> gp.set_real_precision(256) # doesn't seem to do anything
>
> def check_unit(N):
> for d in range(10,N):
> if not is_squarefree(d):
> continue
> K.<a> = QuadraticField(d)
> G = K.unit_group()
> [x,y] = G.gen(1).value()
> x = abs(x)
> R = K.regulator(None)
> twox = round(exp(R))
> x2 = twox/2
> y2 = round(twox/sqrt(d))/2
> print(d,x,x2,y,y2,exp(R)/2)
> if x != x2 or y != y2:
> print("oops!")
> return
> if norm_is_negative(x,d):
> print("norm is negative")
>
--
You received this message because you are subscribed to the Google Groups
"sage-support" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to sage-support+unsubscr...@googlegroups.com.
To view this discussion on the web visit
https://groups.google.com/d/msgid/sage-support/3ffe812a-75ec-4804-ac67-6ffc30f27357n%40googlegroups.com.
