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-rwxr-xr-xmisc_code/2_division.sage126
-rwxr-xr-xmisc_code/2_division_counterex.sage59
-rwxr-xr-xmisc_code/2_division_onecurve.sage101
-rwxr-xr-xmisc_code/2_division_withparisplit.sage122
-rwxr-xr-xmisc_code/Q49-49a1-49a2.gp14
-rwxr-xr-xmisc_code/adelic_failure_example3
-rwxr-xr-xmisc_code/adelic_failure_example.sage10
-rwxr-xr-xmisc_code/ugly_computations/GL2_ZN.py24
-rwxr-xr-xmisc_code/ugly_computations/cm/cm-rank+-all.sage3361
-rwxr-xr-xmisc_code/ugly_computations/cm/test-cm-rank.sage16
-rwxr-xr-xmisc_code/ugly_computations/order_GLn.pdf2137
-rwxr-xr-xmisc_code/ugly_computations/quadratic_class_number.sage12
12 files changed, 5985 insertions, 0 deletions
diff --git a/misc_code/2_division.sage b/misc_code/2_division.sage
new file mode 100755
index 0000000..68cea80
--- /dev/null
+++ b/misc_code/2_division.sage
@@ -0,0 +1,126 @@
1
2R1.<x> = PolynomialRing(QQ)
3R2.<x,y> = PolynomialRing(QQ)
4
5def extended_field( f, A, B, deg_mult ):
6 # f: a polynomial whose roots are the x-coordinates of some points of an
7 # elliptic curve E: y^2 = x^3 + Ax + B.
8 # return value: a field containing the x and y coordinates of those points
9 # deg_mult: a positive integer known to be a multiple of the degree of the
10 # extended field.
11 #
12 # This function uses the properties of resultants (I can provide a pdf
13 # explaining how it works).
14 #
15 # It is useful to compute, e.g., the fields obtained by adjoining the
16 # coordinates of the n-division points of a point (using the n-uplication
17 # formulas to get the required polynomials).
18 #
19 # When used to compute the 2-division fields, it gives the same output as
20 # E.division_field(2).
21
22
23 g = y^2 - x^3 - A*x - B
24 res = f.resultant(g,x)
25 res = res.subs(y=x)
26
27 #K.<b> = f.splitting_field()
28 #print aux
29 print "+++ Computing splitting field of the following: +++"
30
31 pol = R1(res*f)
32 print pol
33
34 K.<b> = NumberField( R( pari(pol).nfsplitting(deg_mult) ) )
35
36 return K
37
38L = []
39
40for A in range(1,9):
41 for B in range(1,9):
42
43 print "Current list of examples:", len(L), "elements. List:"
44 print L
45
46 E = EllipticCurve([0,0,0,A,B])
47 print "*************************"
48 print E
49 print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
50 print "CM:", E.has_cm()
51
52 rep = E.galois_representation()
53 print "mod 2 rep is surjective:", rep.is_surjective(2)
54
55 if E.rank() == 0:
56 print "Stopping because rank 0"
57 print ""
58 continue
59 if len(E.gens()) == 0:
60 print "Stopping because no points of infinite order found"
61 print ""
62 continue
63 if E.has_cm():
64 print "Stopping because CM"
65 print ""
66 print ""
67 continue
68 if not rep.is_surjective(2):
69 print "Stopping because mod 2 rep is not surjective"
70 print ""
71 continue
72
73 K_2.<a> = E.division_field(2)
74 print "2-division field:", K_2
75 P = E(0)
76 flag = False
77 for P in E.gens():
78 if len(P.division_points(2)) == 0:
79 flag = True
80 break
81 if not flag:
82 print "Stopping because the points found are 2-divisible"
83 print ""
84 continue
85 print "Taking the 2-division of P =", P
86
87 # The following polynomial is derived from the duplication formula
88 # (Silverman, p.54) using the x-coordinate of the 2-division point as
89 # an indeterminate.x
90 f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
91 #print f_P.roots()
92
93 M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
94 print "2-division field of P:", M
95
96 if M.degree() != 24:
97 print "Stopping because 2-division of P is too small"
98 print ""
99 continue
100
101 div_pol_4 = E.division_polynomial(4)
102
103 #if div_pol_4.splitting_field('zz').degree() < 48:
104 # print "Stopping because splitting field of div_pol_4 < 48"
105 # exit()
106
107 K_4 = extended_field( div_pol_4, A, B, 96 )
108 print "4-division field:", K_4
109
110 if K_4.degree() != 96:
111 print "Stopping because mod 4 representation not surjective"
112 print ""
113 continue
114
115 if len(f_P.roots(ring=K_4)) == 0:
116 print "Stopping because M is not contained in K_4"
117 print ""
118 continue
119
120 print "----------------"
121 print "|Example Found!|"
122 print "----------------"
123 L.append((A,B))
124
125 print "*************************"
126 print ""
diff --git a/misc_code/2_division_counterex.sage b/misc_code/2_division_counterex.sage
new file mode 100755
index 0000000..f1193fb
--- /dev/null
+++ b/misc_code/2_division_counterex.sage
@@ -0,0 +1,59 @@
1
2R1.<x> = PolynomialRing(QQ)
3R2.<x,y> = PolynomialRing(QQ)
4
5def extended_field( f, A, B, deg_mult ):
6 # f: a polynomial whose roots are the x-coordinates of some points of an
7 # elliptic curve E: y^2 = x^3 + Ax + B.
8 # return value: a field containing the x and y coordinates of those points
9 # deg_mult: a positive integer known to be a multiple of the degree of the
10 # extended field.
11 #
12 # This function uses the properties of resultants (I can provide a pdf
13 # explaining how it works).
14 #
15 # It is useful to compute, e.g., the fields obtained by adjoining the
16 # coordinates of the n-division points of a point (using the n-uplication
17 # formulas to get the required polynomials).
18 #
19 # When used to compute the 2-division fields, it gives the same output as
20 # E.division_field(2).
21
22
23 g = y^2 - x^3 - A*x - B
24 res = f.resultant(g,x)
25 res = res.subs(y=x)
26
27 #print "Splitting field of division pol:"
28 #print R1(f).splitting_field('r')
29 K.<b> = (R1(res*f)).splitting_field(degree_multiple=deg_mult)
30
31 return K
32
33
34A = -3
35B = 17/4
36
37E = EllipticCurve([0,0,0,A,B])
38print E
39print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
40print "CM:", E.has_cm()
41
42rep = E.galois_representation()
43print "mod 2 rep is surjective:", rep.is_surjective(2)
44
45K_2.<a> = E.division_field(2)
46print "2-division field:", K_2
47P = [4,15/2]
48
49# The following polynomial is derived from the duplication formula
50# (Silverman, p.54) using the x-coordinate of the 2-division point as
51# an indeterminate.x
52f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
53
54M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
55print "2-division field of P:", M
56
57if M.degree() != 24:
58 print "Stopping because 2-division of P is too small"
59 exit()
diff --git a/misc_code/2_division_onecurve.sage b/misc_code/2_division_onecurve.sage
new file mode 100755
index 0000000..9cb7043
--- /dev/null
+++ b/misc_code/2_division_onecurve.sage
@@ -0,0 +1,101 @@
1
2R1.<x> = PolynomialRing(QQ)
3R2.<x,y> = PolynomialRing(QQ)
4
5def extended_field( f, A, B, deg_mult ):
6 # f: a polynomial whose roots are the x-coordinates of some points of an
7 # elliptic curve E: y^2 = x^3 + Ax + B.
8 # return value: a field containing the x and y coordinates of those points
9 # deg_mult: a positive integer known to be a multiple of the degree of the
10 # extended field.
11 #
12 # This function uses the properties of resultants (I can provide a pdf
13 # explaining how it works).
14 #
15 # It is useful to compute, e.g., the fields obtained by adjoining the
16 # coordinates of the n-division points of a point (using the n-uplication
17 # formulas to get the required polynomials).
18 #
19 # When used to compute the 2-division fields, it gives the same output as
20 # E.division_field(2).
21
22
23 g = y^2 - x^3 - A*x - B
24 res = f.resultant(g,x)
25 res = res.subs(y=x)
26
27 #print "Splitting field of division pol:"
28 #print R1(f).splitting_field('r')
29 K.<b> = (R1(res*f)).splitting_field(degree_multiple=deg_mult)
30
31 return K
32
33
34A = -3
35B = 17/4
36
37E = EllipticCurve([0,0,0,A,B])
38print "*************************"
39print E
40print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
41print "CM:", E.has_cm()
42
43rep = E.galois_representation()
44print "mod 2 rep is surjective:", rep.is_surjective(2)
45
46if E.rank() == 0:
47 print "Stopping because rank 0"
48 exit()
49if len(E.gens()) == 0:
50 print "Stopping because no points of infinite order found"
51 exit()
52if E.has_cm():
53 print "Stopping because CM"
54 print ""
55 exit()
56if not rep.is_surjective(2):
57 print "Stopping because mod 2 rep is not surjective"
58 exit()
59
60K_2.<a> = E.division_field(2)
61print "2-division field:", K_2
62P = E(0)
63flag = False
64for P in E.gens():
65 if len(P.division_points(2)) == 0:
66 flag = True
67 break
68if not flag:
69 print "Stopping because the points found are 2-divisible"
70 exit()
71print "Taking the 2-division of P =", P
72
73# The following polynomial is derived from the duplication formula
74# (Silverman, p.54) using the x-coordinate of the 2-division point as
75# an indeterminate.x
76f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
77
78M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
79print "2-division field of P:", M
80
81if M.degree() != 24:
82 print "Stopping because 2-division of P is too small"
83 exit()
84
85K_4 = extended_field( E.division_polynomial(4), A, B, 96 )
86print "4-division field:", K_4
87
88if K_4.degree() != 96:
89 print "Stopping because mod 4 representation not surjective"
90 exit()
91
92if len(f_P.roots(ring=K_4)) == 0:
93 print "Stopping because E is not contained in K_4"
94 exit()
95
96print "----------------"
97print "|Example Found!|"
98print "----------------"
99
100print "*************************"
101print ""
diff --git a/misc_code/2_division_withparisplit.sage b/misc_code/2_division_withparisplit.sage
new file mode 100755
index 0000000..349ea95
--- /dev/null
+++ b/misc_code/2_division_withparisplit.sage
@@ -0,0 +1,122 @@
1
2R1.<x> = PolynomialRing(QQ)
3R2.<x,y> = PolynomialRing(QQ)
4
5def extended_field( f, A, B, deg_mult ):
6 # f: a polynomial whose roots are the x-coordinates of some points of an
7 # elliptic curve E: y^2 = x^3 + Ax + B.
8 # return value: a field containing the x and y coordinates of those points
9 # deg_mult: a positive integer known to be a multiple of the degree of the
10 # extended field.
11 #
12 # This function uses the properties of resultants (I can provide a pdf
13 # explaining how it works).
14 #
15 # It is useful to compute, e.g., the fields obtained by adjoining the
16 # coordinates of the n-division points of a point (using the n-uplication
17 # formulas to get the required polynomials).
18 #
19 # When used to compute the 2-division fields, it gives the same output as
20 # E.division_field(2).
21
22
23 g = y^2 - x^3 - A*x - B
24 res = f.resultant(g,x)
25 res = res.subs(y=x)
26
27 #K.<b> = f.splitting_field()
28 #print aux
29 pol = pari(R1(res*f)).nfsplitting(deg_mult)
30 K.<b> = NumberField(g)
31
32 return K
33
34L = []
35
36for A in range(1,9):
37 for B in range(1,9):
38
39 print "Current list of examples:", len(L), "elements. List:"
40 print L
41
42 E = EllipticCurve([0,0,0,A,B])
43 print "*************************"
44 print E
45 print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
46 print "CM:", E.has_cm()
47
48 rep = E.galois_representation()
49 print "mod 2 rep is surjective:", rep.is_surjective(2)
50
51 if E.rank() == 0:
52 print "Stopping because rank 0"
53 print ""
54 continue
55 if len(E.gens()) == 0:
56 print "Stopping because no points of infinite order found"
57 print ""
58 continue
59 if E.has_cm():
60 print "Stopping because CM"
61 print ""
62 print ""
63 continue
64 if not rep.is_surjective(2):
65 print "Stopping because mod 2 rep is not surjective"
66 print ""
67 continue
68
69 K_2.<a> = E.division_field(2)
70 print "2-division field:", K_2
71 P = E(0)
72 flag = False
73 for P in E.gens():
74 if len(P.division_points(2)) == 0:
75 flag = True
76 break
77 if not flag:
78 print "Stopping because the points found are 2-divisible"
79 print ""
80 continue
81 print "Taking the 2-division of P =", P
82
83 # The following polynomial is derived from the duplication formula
84 # (Silverman, p.54) using the x-coordinate of the 2-division point as
85 # an indeterminate.x
86 f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
87 #print f_P.roots()
88
89 M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
90 print "2-division field of P:", M
91
92 if M.degree() != 24:
93 print "Stopping because 2-division of P is too small"
94 print ""
95 continue
96
97 div_pol_4 = E.division_polynomial(4)
98
99 #if div_pol_4.splitting_field('zz').degree() < 48:
100 # print "Stopping because splitting field of div_pol_4 < 48"
101 # exit()
102
103 K_4 = extended_field( div_pol_4, A, B, 96 )
104 print "4-division field:", K_4
105
106 if K_4.degree() != 96:
107 print "Stopping because mod 4 representation not surjective"
108 print ""
109 continue
110
111 if len(f_P.roots(ring=K_4)) == 0:
112 print "Stopping because M is not contained in K_4"
113 print ""
114 continue
115
116 print "----------------"
117 print "|Example Found!|"
118 print "----------------"
119 L.append((A,B))
120
121 print "*************************"
122 print ""
diff --git a/misc_code/Q49-49a1-49a2.gp b/misc_code/Q49-49a1-49a2.gp
new file mode 100755
index 0000000..a8ae14d
--- /dev/null
+++ b/misc_code/Q49-49a1-49a2.gp
@@ -0,0 +1,14 @@
1elltorsfield_deg( E, n, deg ) = {
2 my( f, eqn, res );
3 f = elldivpol( E, n );
4 eqn = y^2 + E[1]*x*y + E[3]*y - x^3 - E[2]*x^2 - E[4]*x - E[5];
5 res = substpol( polresultant( f, eqn ), y, x );
6 printf( "Computing %d-torsion field...\n", n );
7 return( nfinit( nfsplitting( f*res, deg ) ) );
8}
9
10print("Computing 49-torsion field for 49a1, expecting maximal degree 42*49\n");
11elltorsfield_deg( ellinit( "49a1" ), 49, 42*49 );
12
13print("Computing 49-torsion field for 49a1, expecting maximal degree 42*49\n");
14elltorsfield_deg( ellinit( "49a1" ), 49, 42*49 );
diff --git a/misc_code/adelic_failure_example b/misc_code/adelic_failure_example
new file mode 100755
index 0000000..d84180e
--- /dev/null
+++ b/misc_code/adelic_failure_example
@@ -0,0 +1,3 @@
1E = EllipticCurve([0, 1, 0, -44, -84])
2print "Rational torsion points:", E.torsion_points()
3print "Generator of the free part:", E.gens()
diff --git a/misc_code/adelic_failure_example.sage b/misc_code/adelic_failure_example.sage
new file mode 100755
index 0000000..1fdca8c
--- /dev/null
+++ b/misc_code/adelic_failure_example.sage
@@ -0,0 +1,10 @@
1E = EllipticCurve([0, 1, 0, -44, -84])
2print "Rational torsion points:", E.torsion_points()
3print "Generators of the free part:", E.gens()
4P = E.gens()[0]
5
6K = CyclotomicField(3)
7EK = E.base_extend(K)
8PK = EK.point(P)
9
10print "2-division points over Q(zeta_3):", PK.division_points(2)
diff --git a/misc_code/ugly_computations/GL2_ZN.py b/misc_code/ugly_computations/GL2_ZN.py
new file mode 100755
index 0000000..c706703
--- /dev/null
+++ b/misc_code/ugly_computations/GL2_ZN.py
@@ -0,0 +1,24 @@
1# Prints a list of matrices, the elements of GL_2(Z/nZ)
2
3def gcd( m, n ):
4 if n == 0:
5 return m
6 return gcd( n, m%n )
7
8def print_mat( a, b, c, d ):
9 print ""
10 print "|", a, " ", b, "|"
11 print "|", c, " ", d, "|"
12
13n = input("Choose n: ")
14
15count = 0
16for a in range(n):
17 for b in range(n):
18 for c in range(n):
19 for d in range(n):
20 if gcd( n, abs(a*d-b*c)) == 1:
21 print_mat(a,b,c,d)
22 count += 1
23
24print "Found", count, "elements"
diff --git a/misc_code/ugly_computations/cm/cm-rank+-all.sage b/misc_code/ugly_computations/cm/cm-rank+-all.sage
new file mode 100755
index 0000000..f1943af
--- /dev/null
+++ b/misc_code/ugly_computations/cm/cm-rank+-all.sage
@@ -0,0 +1,3361 @@
1
2# Elliptic curves downloaded from the LMFDB downloaded on 23 October 2019.
3# Below is a list called data. Each entry has the form:
4# [a1,a2,a3,a4,a6] (Weierstrass Coefficients)
5
6
7data = [\
8[0, -1, 1, -887, -10143],\
9[0, -1, 1, -7, 10],\
10[0, 0, 1, 0, -34],\
11[0, 0, 1, 0, 1],\
12[0, 0, 1, 0, -1],\
13[0, 0, 1, 0, 20],\
14[0, 1, 0, -13, -21],\
15[0, 1, 0, -3, 1],\
16[0, 0, 0, -2, 0],\
17[0, 0, 0, 8, 0],\
18[0, 0, 0, -12, 0],\
19[0, 0, 0, 3, 0],\
20[0, 0, 1, -13718, -619025],\
21[0, 0, 1, -38, 90],\
22[0, 0, 0, 0, -4],\
23[0, 0, 0, 0, 108],\
24[1, -1, 1, -16400, -804212],\
25[1, -1, 1, -965, -13940],\
26[1, -1, 1, -335, 2440],\
27[1, -1, 1, -20, 46],\
28[0, 0, 1, 0, -331],\
29[0, 0, 1, 0, 12],\
30[0, 0, 0, -396, -3024],\
31[0, 0, 0, -396, 3024],\
32[0, 0, 0, -36, 0],\
33[0, 0, 0, 9, 0],\
34[0, 0, 0, -540, -4752],\
35[0, 0, 0, -60, 176],\
36[0, 0, 0, 0, -216],\
37[0, 0, 0, 0, 8],\
38[0, 0, 1, -6750, -213469],\
39[0, 0, 1, -750, 7906],\
40[0, 0, 1, 0, -844],\
41[0, 0, 1, 0, 31],\
42[0, 0, 0, -29155, -1915998],\
43[0, 0, 0, -1715, -33614],\
44[0, 0, 0, -595, 5586],\
45[0, 0, 0, -35, 98],\
46[0, 0, 0, -275, -1750],\
47[0, 0, 0, -275, 1750],\
48[0, 0, 0, -25, 0],\
49[0, 0, 0, 100, 0],\
50[0, 0, 0, -5, 0],\
51[0, 0, 0, 20, 0],\
52[0, 0, 0, 0, -2700],\
53[0, 0, 0, 0, 100],\
54[0, 0, 0, 0, -972],\
55[0, 0, 0, 0, 36],\
56[0, 0, 0, 0, -243],\
57[0, 0, 0, 0, 9],\
58[0, 0, 1, 0, -2],\
59[0, 0, 1, 0, 47],\
60[0, 0, 1, 0, -16207],\
61[0, 0, 1, 0, 600],\
62[0, 0, 1, 0, -192787],\
63[0, 0, 1, 0, 7140],\
64[0, 0, 1, 0, -88],\
65[0, 0, 1, 0, 3],\
66[0, 0, 0, -539, -4802],\
67[0, 0, 0, -539, 4802],\
68[0, 0, 0, -49, 0],\
69[0, 0, 0, 196, 0],\
70[0, 0, 0, -500, 0],\
71[0, 0, 0, 125, 0],\
72[0, 0, 0, -20, 0],\
73[0, 0, 0, 5, 0],\
74[0, 0, 0, -1080, -13662],\
75[0, 0, 0, -120, 506],\
76[0, 0, 0, 0, -54],\
77[0, 0, 0, 0, 2],\
78[0, 0, 0, -1080, 13662],\
79[0, 0, 0, -120, -506],\
80[0, 0, 0, 0, -2],\
81[0, 0, 0, 0, 54],\
82[0, 0, 0, 0, -28],\
83[0, 0, 0, 0, 756],\
84[0, 0, 0, -6615, 203742],\
85[0, 0, 0, -735, -7546],\
86[0, 0, 0, 0, -343],\
87[0, 0, 0, 0, 9261],\
88[0, 0, 1, -1590140, -771794326],\
89[0, 0, 1, -860, 9707],\
90[0, 0, 0, -72, 0],\
91[0, 0, 0, 18, 0],\
92[0, 0, 0, -6, 0],\
93[0, 0, 0, 24, 0],\
94[0, 0, 0, -120, -448],\
95[0, 0, 0, -30, 56],\
96[0, 0, 0, -54, 0],\
97[0, 0, 0, 216, 0],\
98[0, 0, 0, 0, -13500],\
99[0, 0, 0, 0, 500],\
100[0, 0, 0, 0, -16875],\
101[0, 0, 0, 0, 625],\
102[0, 0, 0, 0, -135],\
103[0, 0, 0, 0, 5],\
104[0, 1, 1, -22183, -1312206],\
105[0, 1, 1, -183, 919],\
106[0, 0, 0, -7, 0],\
107[0, 0, 0, 28, 0],\
108[0, 0, 0, -2156, -38416],\
109[0, 0, 0, -2156, 38416],\
110[0, 0, 0, -196, 0],\
111[0, 0, 0, 49, 0],\
112[0, 0, 0, -116620, -15327984],\
113[0, 0, 0, -6860, -268912],\
114[0, 0, 0, -2380, 44688],\
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diff --git a/misc_code/ugly_computations/cm/test-cm-rank.sage b/misc_code/ugly_computations/cm/test-cm-rank.sage
new file mode 100755
index 0000000..7671838
--- /dev/null
+++ b/misc_code/ugly_computations/cm/test-cm-rank.sage
@@ -0,0 +1,16 @@
1def test():
2 attach("cm-rank+-all.sage")
3 no = 0
4 unk = 0
5 for cf in data:
6 E = EllipticCurve(cf)
7 E_K = E.base_extend(NumberField(x^2-E.cm_discriminant(),'a'))
8 try:
9 if E_K.rank() != 2* E.rank():
10 print "Non doubling:", cf, E.rank(), E_K.rank()
11 no += 1
12 except:
13 print "Can't compute the rank for", cf
14 unk += 1
15 print "Total ok curves:", (len(data)-(no+unk)), "/", len(data), \
16 "( unknown:", unk, ")"
diff --git a/misc_code/ugly_computations/order_GLn.pdf b/misc_code/ugly_computations/order_GLn.pdf
new file mode 100755
index 0000000..9449f3a
--- /dev/null
+++ b/misc_code/ugly_computations/order_GLn.pdf
@@ -0,0 +1,2137 @@
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diff --git a/misc_code/ugly_computations/quadratic_class_number.sage b/misc_code/ugly_computations/quadratic_class_number.sage
new file mode 100755
index 0000000..a26dc50
--- /dev/null
+++ b/misc_code/ugly_computations/quadratic_class_number.sage
@@ -0,0 +1,12 @@
1# Determines the proportion of real quadratic fields with odd class number.
2def f(B):
3 sqf = [ b for b in range(2,B) if b%4==1 and is_squarefree(b) ]
4 total_odd = 0
5
6 R.<x> = PolynomialRing(QQ)
7 for a in sqf:
8 if NumberField( x^2 - a, 'c' ).class_number() % 2 == 1:
9 total_odd += 1
10
11 print "=1 mod 4 squarefree integers up to", B, ": ", len(sqf)
12 print "With odd class number:", total_odd

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