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### compute regulator with more precision

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. = 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")

### compute regulator with more precision

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 = 331.331. Sorry, I don't know how to get the code to display with newlines and tabs correct. It looks right when I edit the post.

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. = 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") negative")

 3 None Emmanuel Charpentier 5416 ●5 ●38 ●111

### compute regulator with more precision

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. Sorry, I don't know how to get the code to display with newlines and tabs correct. It looks right when I edit the post.

gp.set_real_precision(256)  # doesn't seem to do anything anything

def check_unit(N):
for d in range(10,N):
if not is_squarefree(d):
continue
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") negative")


### compute regulator with more precision

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. Sorry, I don't know how to get the code to display with newlines and tabs correct. It looks right when I edit the post.

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
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")


A shorter "minimal non-working example":

sage: d = 2352721672669
sage: gp.default("realprecision",256)
0 sage: R = K.regulator()

followed by many error messages.

### compute regulator with more precision

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. Sorry, I don't know how to get the code to display with newlines and tabs correct. It looks right when I edit the post.

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
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")


A shorter "minimal non-working example":

 sage: d = 2352721672669
sage: gp.default("realprecision",256)
0 sage: R = K.regulator()
''

followed by many error messages.

### compute regulator with more precision

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. Sorry, I don't know how to get the code to display with newlines and tabs correct. It looks right when I edit the post.

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
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")


A shorter "minimal non-working example":

 sage: d = 2352721672669
K.\<a\&gt; =="" quadraticfield(d)="" <br=""> sage: gp.default("realprecision",256)
0 sage: R = K.regulator()
''

followed by many error messages.

### compute regulator with more precision

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. Sorry, I don't know how to get the code to display with newlines and tabs correct. It looks right when I edit the post.

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
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")


A shorter "minimal non-working example":

sage: d = 2352721672669
sage: gp.default("realprecision",256)
0 sage: R = K.regulator()

followed by many error messages.

### compute regulator with more precision

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. Sorry, I don't know how to get the code to display with newlines and tabs correct. It looks right when I edit the post.

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
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")


A shorter "minimal non-working example":

sage: d = 2352721672669
sage: gp.default("realprecision",256)
0
sage: R = K.regulator()   K.regulator()


followed by many error messages.

### compute regulator with more precision

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. Sorry, I don't know how to get the code to display with newlines and tabs correct. It looks right when I edit the post.

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
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")


A shorter "minimal non-working example":

sage: d = 2352721672669