You could just do R =
K.embeddings(RealField(1000))[0](K.units()[0]).log().abs(), for instance.
On Sunday, December 26, 2021 at 9:21:18 PM UTC+1 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")
>
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