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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