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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")

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")

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
                K. 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")
 negative")

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
                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")

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
                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")

A shorter "minimal non-working example":

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

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
                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")

A shorter "minimal non-working example":

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

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
                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")

A shorter "minimal non-working example":

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

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
                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")

A shorter "minimal non-working example":

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

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
                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")

A shorter "minimal non-working example":

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

followed by many error messages.