Weil q-number

799 days ago by wangwh

x = PolynomialRing(QQ, 'x').gen() K.<a>=NumberField(x^4+x^3+x^2+x+1) K 
        
Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1
Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1
K.factor(61) 
        
(Fractional ideal (-a^3 + 2*a^2 - a - 1)) * (Fractional ideal (2*a^3 -
a^2 - a - 1)) * (Fractional ideal (-3*a^3 - a^2)) * (Fractional ideal
(-3*a^2 - 1))
(Fractional ideal (-a^3 + 2*a^2 - a - 1)) * (Fractional ideal (2*a^3 - a^2 - a - 1)) * (Fractional ideal (-3*a^3 - a^2)) * (Fractional ideal (-3*a^2 - 1))
(a+a^4).minpoly().disc().factor() 
       
 
       
L.<t>=NumberField(x^2+x-1) L.factor(61) 
        
(Fractional ideal (7*t + 4)) * (Fractional ideal (7*t + 3))
(Fractional ideal (7*t + 4)) * (Fractional ideal (7*t + 3))
K.factor(7*a+7*a^4+4) 
        
(Fractional ideal (-a^3 + 2*a^2 - a - 1)) * (Fractional ideal (-3*a^2 -
1))
(Fractional ideal (-a^3 + 2*a^2 - a - 1)) * (Fractional ideal (-3*a^2 - 1))
b1=-a^3 + 2*a^2 - a - 1 
       
b4=-3*a^2 -1 
       
b2=2*a^3 -a^2 - a - 1 
       
b3=-3*a^3 - a^2 
       
p1=b1*b3 p1 
        
3*a^3 + 2*a - 6
3*a^3 + 2*a - 6
minpoly(p1).disc().factor() 
        
5^3 * 61^2
5^3 * 61^2
minpoly(a*p1).disc().factor() 
        
5^9 * 19^2 * 61^2
5^9 * 19^2 * 61^2
minpoly(a^2*p1).disc().factor() 
        
2^20 * 5^3 * 61^2
2^20 * 5^3 * 61^2
minpoly(a^3*p1).disc().factor() 
        
5^3 * 11^4 * 41^2 * 61^2
5^3 * 11^4 * 41^2 * 61^2
minpoly(a^4*p1).disc().factor() 
        
3^12 * 5^3 * 61^2
3^12 * 5^3 * 61^2
minpoly(b1).disc().factor() 
        
5^7 * 11^2
5^7 * 11^2
minpoly(a*b1).disc().factor() 
        
3^12 * 5^3
3^12 * 5^3
minpoly(a^2*b1).disc().factor() 
        
2^4 * 5^3 * 11^4
2^4 * 5^3 * 11^4
minpoly(a^3*b1).disc().factor() 
        
5^3
5^3
minpoly(a^4*b1).disc().factor() 
        
5^3 * 11^2 * 31^2
5^3 * 11^2 * 31^2
minpoly(b2).disc() 
        
14535125
14535125
minpoly(a*b2).disc().factor() 
        
3^12 * 5^3
3^12 * 5^3
minpoly(a^2*b2).disc().factor() 
        
5^3
5^3
minpoly(b3).disc().factor() 
        
2^4 * 5^3 * 11^4
2^4 * 5^3 * 11^4
minpoly(b4).disc().factor() 
        
3^12 * 5^3
3^12 * 5^3
a^3*b1 
        
a^2 + 3
a^2 + 3
minpoly(a^3*b1) 
        
x^4 - 11*x^3 + 46*x^2 - 86*x + 61
x^4 - 11*x^3 + 46*x^2 - 86*x + 61
minpoly(p1) 
        
x^4 + 29*x^3 + 331*x^2 + 1769*x + 3721
x^4 + 29*x^3 + 331*x^2 + 1769*x + 3721
minpoly(1-p1) 
        
x^4 - 33*x^3 + 424*x^2 - 2522*x + 5851
x^4 - 33*x^3 + 424*x^2 - 2522*x + 5851
minpoly(1+p1) 
        
x^4 + 25*x^3 + 250*x^2 + 1190*x + 2255
x^4 + 25*x^3 + 250*x^2 + 1190*x + 2255
is_prime(5851) 
        
True
True
a^3*p1 
        
-8*a^3 - 2*a^2 + a - 2
-8*a^3 - 2*a^2 + a - 2
a^4*b1 
        
a^3 + 3*a
a^3 + 3*a
is_prime(311) 
        
True
True
K.factor(311) 
        
(Fractional ideal (5*a^3 + 3*a^2 + 2*a + 1)) * (Fractional ideal (2*a^3
+ 5*a^2 + a + 3)) * (Fractional ideal (2*a^3 - 3*a^2 - 2*a - 1)) *
(Fractional ideal (2*a^3 - 2*a^2 + a + 3))
(Fractional ideal (5*a^3 + 3*a^2 + 2*a + 1)) * (Fractional ideal (2*a^3 + 5*a^2 + a + 3)) * (Fractional ideal (2*a^3 - 3*a^2 - 2*a - 1)) * (Fractional ideal (2*a^3 - 2*a^2 + a + 3))
L.factor(311) 
        
(Fractional ideal (16*t + 5)) * (Fractional ideal (16*t + 11))
(Fractional ideal (16*t + 5)) * (Fractional ideal (16*t + 11))
K.factor(16*a+16*a^4+5) 
        
(Fractional ideal (5*a^3 + 3*a^2 + 2*a + 1)) * (Fractional ideal (2*a^3
- 3*a^2 - 2*a - 1))
(Fractional ideal (5*a^3 + 3*a^2 + 2*a + 1)) * (Fractional ideal (2*a^3 - 3*a^2 - 2*a - 1))
p2=(5*a^3 + 3*a^2 + 2*a + 1)* (2*a^3+ 5*a^2 + a + 3) 
       
minpoly(p2).disc().factor() 
        
5^9 * 11^2 * 331^2
5^9 * 11^2 * 331^2
minpoly(p2^5).disc().factor() 
        
2^12 * 3^4 * 5^21 * 7^4 * 11^12 * 19^4 * 31^4 * 199^2 * 311^10
2^12 * 3^4 * 5^21 * 7^4 * 11^12 * 19^4 * 31^4 * 199^2 * 311^10
minpoly(a*p2).disc().factor() 
        
2^12 * 5^3 * 89^2 * 331^2
2^12 * 5^3 * 89^2 * 331^2
minpoly(a^2*p2).disc().factor() 
        
5^3 * 19^4 * 109^2 * 331^2
5^3 * 19^4 * 109^2 * 331^2
minpoly(a^3*p2).disc().factor() 
        
3^12 * 5^3 * 59^2 * 331^2
3^12 * 5^3 * 59^2 * 331^2
k=FiniteField(29^2,'c') g=k.gen() 
       
h=g^5 
       
for i in [0..71]: for j in [0..71]: if i<j: q1=h^i q2=h^j q3=q1^19 q4=q2^19 print i, j, Mod((q1-q2)*(q1-q3)*(q1-q4)*(q2-q3)*(q2-q4)*(q3-q4),19) 
        
WARNING: Output truncated!  
full_output.txt



0 1 0
0 2 0
0 3 0
0 4 0
0 5 0
0 6 0
0 7 0
0 8 0
0 9 0
0 10 0
0 11 0
0 12 0
0 13 0
0 14 0
0 15 0
0 16 0
0 17 0
0 18 0
0 19 0
0 20 0
0 21 0
0 22 0
0 23 0
0 24 0
0 25 0
0 26 0
0 27 0
0 28 0
0 29 0
0 30 0
0 31 0
0 32 0
0 33 0
0 34 0
0 35 0
0 36 0
0 37 0
0 38 0
0 39 0
0 40 0
0 41 0
0 42 0
0 43 0
0 44 0
0 45 0
0 46 0
0 47 0
0 48 0
0 49 0
0 50 0
0 51 0
0 52 0
0 53 0
0 54 0
0 55 0
0 56 0
0 57 0
0 58 0
0 59 0

...

60 67 0
60 68 0
60 69 0
60 70 0
60 71 0
61 62 2
61 63 9
61 64 0
61 65 12
61 66 10
61 67 14
61 68 0
61 69 1
61 70 3
61 71 12
62 63 5
62 64 0
62 65 1
62 66 13
62 67 6
62 68 0
62 69 1
62 70 4
62 71 17
63 64 0
63 65 18
63 66 12
63 67 18
63 68 0
63 69 6
63 70 12
63 71 11
64 65 0
64 66 0
64 67 0
64 68 0
64 69 0
64 70 0
64 71 0
65 66 14
65 67 6
65 68 0
65 69 8
65 70 13
65 71 3
66 67 16
66 68 0
66 69 7
66 70 15
66 71 4
67 68 0
67 69 12
67 70 8
67 71 12
68 69 0
68 70 0
68 71 0
69 70 3
69 71 4
70 71 17
WARNING: Output truncated!  
full_output.txt



0 1 0
0 2 0
0 3 0
0 4 0
0 5 0
0 6 0
0 7 0
0 8 0
0 9 0
0 10 0
0 11 0
0 12 0
0 13 0
0 14 0
0 15 0
0 16 0
0 17 0
0 18 0
0 19 0
0 20 0
0 21 0
0 22 0
0 23 0
0 24 0
0 25 0
0 26 0
0 27 0
0 28 0
0 29 0
0 30 0
0 31 0
0 32 0
0 33 0
0 34 0
0 35 0
0 36 0
0 37 0
0 38 0
0 39 0
0 40 0
0 41 0
0 42 0
0 43 0
0 44 0
0 45 0
0 46 0
0 47 0
0 48 0
0 49 0
0 50 0
0 51 0
0 52 0
0 53 0
0 54 0
0 55 0
0 56 0
0 57 0
0 58 0
0 59 0

...

60 67 0
60 68 0
60 69 0
60 70 0
60 71 0
61 62 2
61 63 9
61 64 0
61 65 12
61 66 10
61 67 14
61 68 0
61 69 1
61 70 3
61 71 12
62 63 5
62 64 0
62 65 1
62 66 13
62 67 6
62 68 0
62 69 1
62 70 4
62 71 17
63 64 0
63 65 18
63 66 12
63 67 18
63 68 0
63 69 6
63 70 12
63 71 11
64 65 0
64 66 0
64 67 0
64 68 0
64 69 0
64 70 0
64 71 0
65 66 14
65 67 6
65 68 0
65 69 8
65 70 13
65 71 3
66 67 16
66 68 0
66 69 7
66 70 15
66 71 4
67 68 0
67 69 12
67 70 8
67 71 12
68 69 0
68 70 0
68 71 0
69 70 3
69 71 4
70 71 17
for i in [0..168]: for j in [0..168]: if i<j: q1=h^i q2=h^j q3=q1^29 q4=q2^29 if Mod(q1*q3,29)==1: if Mod(q2*q4,29)==1: print i, j, Mod((q1-q2)*(q1-q3)*(q1-q4)*(q2-q3)*(q2-q4)*(q3-q4),29) 
        
0 28 0
0 56 0
0 84 0
0 112 0
0 140 0
0 168 0
28 56 12
28 84 0
28 112 17
28 140 0
28 168 0
56 84 0
56 112 0
56 140 17
56 168 0
84 112 0
84 140 0
84 168 0
112 140 12
112 168 0
140 168 0
0 28 0
0 56 0
0 84 0
0 112 0
0 140 0
0 168 0
28 56 12
28 84 0
28 112 17
28 140 0
28 168 0
56 84 0
56 112 0
56 140 17
56 168 0
84 112 0
84 140 0
84 168 0
112 140 12
112 168 0
140 168 0
Mod(17,29)^4 
        
1
1
 
        
3
3
Mod(3,31)^5 
        
26
26
for i in [1..5]: print Mod(26,31)^i 
        
26
25
30
5
6
26
25
30
5
6
Mod(2*8*12*6*14*4,31)^2 
        
1
1
is_prime(1861) 
        
True
True
K.factor(1861) 
        
(Fractional ideal (7*a^3 + 7*a^2 + 3*a + 4)) * (Fractional ideal (-4*a^3
- 7*a^2 - 3)) * (Fractional ideal (7*a^3 + 4*a^2 + 3)) * (Fractional
ideal (-4*a^3 - 4*a^2 + 3*a - 1))
(Fractional ideal (7*a^3 + 7*a^2 + 3*a + 4)) * (Fractional ideal (-4*a^3 - 7*a^2 - 3)) * (Fractional ideal (7*a^3 + 4*a^2 + 3)) * (Fractional ideal (-4*a^3 - 4*a^2 + 3*a - 1))
L.factor(1861) 
        
(Fractional ideal (5*t + 46)) * (Fractional ideal (-5*t + 41))
(Fractional ideal (5*t + 46)) * (Fractional ideal (-5*t + 41))
K.factor(5*a+5*a^4+46) 
        
(Fractional ideal (7*a^3 + 7*a^2 + 3*a + 4)) * (Fractional ideal (-4*a^3
- 4*a^2 + 3*a - 1))
(Fractional ideal (7*a^3 + 7*a^2 + 3*a + 4)) * (Fractional ideal (-4*a^3 - 4*a^2 + 3*a - 1))
p31=(7*a^3 + 7*a^2 + 3*a + 4)*(-4*a^3- 7*a^2 - 3) 
       
minpoly(p31^5).disc().factor() 
        
2^24 * 3^12 * 5^21 * 7^4 * 11^4 * 19^4 * 29^2 * 31^2 * 41^2 * 61^4 *
199^2 * 1471^2 * 1861^10
2^24 * 3^12 * 5^21 * 7^4 * 11^4 * 19^4 * 29^2 * 31^2 * 41^2 * 61^4 * 199^2 * 1471^2 * 1861^10
Mod(221096161914265125,31) 
        
0
0
for i in [1..100]: q=310*i+1 if is_prime(q)==True: print q 
        
311
1861
2791
4651
5581
8681
11161
11471
12401
13331
16741
17981
18911
19531
19841
20771
21391
21701
23251
23561
24181
25111
26041
27281
27901
28211
29761
30071
311
1861
2791
4651
5581
8681
11161
11471
12401
13331
16741
17981
18911
19531
19841
20771
21391
21701
23251
23561
24181
25111
26041
27281
27901
28211
29761
30071
K.factor(2791) 
        
(Fractional ideal (6*a^3 + 6*a^2 - a + 2)) * (Fractional ideal (-6*a^3 -
7*a^2 - 4)) * (Fractional ideal (7*a^3 + 7*a^2 + a + 3)) * (Fractional
ideal (-7*a^3 - 6*a^2 - 4))
(Fractional ideal (6*a^3 + 6*a^2 - a + 2)) * (Fractional ideal (-6*a^3 - 7*a^2 - 4)) * (Fractional ideal (7*a^3 + 7*a^2 + a + 3)) * (Fractional ideal (-7*a^3 - 6*a^2 - 4))
L.factor(2791) 
        
(Fractional ideal (10*t - 49)) * (Fractional ideal (-10*t - 59))
(Fractional ideal (10*t - 49)) * (Fractional ideal (-10*t - 59))
K.factor(10*a+10*a^4-49) 
        
(Fractional ideal (-6*a^3 - 7*a^2 - 4)) * (Fractional ideal (-7*a^3 -
6*a^2 - 4))
(Fractional ideal (-6*a^3 - 7*a^2 - 4)) * (Fractional ideal (-7*a^3 - 6*a^2 - 4))
p2791=(-6*a^3 - 7*a^2 - 4)*(6*a^3 + 6*a^2 - a + 2) 
       
minpoly(p2791^5).disc().factor() 
        
2^12 * 3^12 * 5^21 * 11^4 * 19^4 * 29^2 * 59^4 * 61^4 * 79^2 * 101^2 *
139^2 * 701^2 * 2791^10
2^12 * 3^12 * 5^21 * 11^4 * 19^4 * 29^2 * 59^4 * 61^4 * 79^2 * 101^2 * 139^2 * 701^2 * 2791^10
Mod(minpoly(p2791^5).disc(),31) 
        
20
20
Mod(20*21*19*20,31)^2 
        
20
20
Mod(20*25*19*24*5,31)^2 
        
5
5
Mod(21*25*20*24*4,31)^2 
        
16
16
Mod(20*21*25*5*4,31)^2 
        
5
5
Mod(19*20*24*5*4,31)^2 
        
16
16
for i in [1,5,6,25,26,30]: for j in [1,5,6,25,26,30]: for k in [1,5,6,25,26,30]: for l in [1,5,6,25,26,30]: if i<j: if j<k: if k<l: print i,j,k,l, "discriminant is congruent to:", Mod((i-j)*(i-k)*(i-l)*(j-k)*(j-l)*(k-l),31)^2 
        
1 5 6 25 discriminant is congruent to: 16
1 5 6 26 discriminant is congruent to: 5
1 5 6 30 discriminant is congruent to: 5
1 5 25 26 discriminant is congruent to: 16
1 5 25 30 discriminant is congruent to: 16
1 5 26 30 discriminant is congruent to: 20
1 6 25 26 discriminant is congruent to: 5
1 6 25 30 discriminant is congruent to: 20
1 6 26 30 discriminant is congruent to: 16
1 25 26 30 discriminant is congruent to: 5
5 6 25 26 discriminant is congruent to: 20
5 6 25 30 discriminant is congruent to: 5
5 6 26 30 discriminant is congruent to: 16
5 25 26 30 discriminant is congruent to: 5
6 25 26 30 discriminant is congruent to: 16
1 5 6 25 discriminant is congruent to: 16
1 5 6 26 discriminant is congruent to: 5
1 5 6 30 discriminant is congruent to: 5
1 5 25 26 discriminant is congruent to: 16
1 5 25 30 discriminant is congruent to: 16
1 5 26 30 discriminant is congruent to: 20
1 6 25 26 discriminant is congruent to: 5
1 6 25 30 discriminant is congruent to: 20
1 6 26 30 discriminant is congruent to: 16
1 25 26 30 discriminant is congruent to: 5
5 6 25 26 discriminant is congruent to: 20
5 6 25 30 discriminant is congruent to: 5
5 6 26 30 discriminant is congruent to: 16
5 25 26 30 discriminant is congruent to: 5
6 25 26 30 discriminant is congruent to: 16
Mod(20,31)^5 
        
25
25
K.factor(11) 
        
(Fractional ideal (-a^3 - 2*a - 1)) * (Fractional ideal (a^3 - a + 1)) *
(Fractional ideal (-2*a^3 - 2*a^2 - a - 1)) * (Fractional ideal (-2*a^3
- a^2 - a))
(Fractional ideal (-a^3 - 2*a - 1)) * (Fractional ideal (a^3 - a + 1)) * (Fractional ideal (-2*a^3 - 2*a^2 - a - 1)) * (Fractional ideal (-2*a^3 - a^2 - a))
L.factor(11) 
        
(Fractional ideal (3*t + 1)) * (Fractional ideal (3*t + 2))
(Fractional ideal (3*t + 1)) * (Fractional ideal (3*t + 2))
K.factor(3*a+3*a^4+1) 
        
(Fractional ideal (-2*a^3 - 2*a^2 - a - 1)) * (Fractional ideal (-2*a^3
- a^2 - a))
(Fractional ideal (-2*a^3 - 2*a^2 - a - 1)) * (Fractional ideal (-2*a^3 - a^2 - a))
(-2*a^3 - 2*a^2 - a - 1)*(-a^3 - 2*a - 1) 
        
2*a^3 - a^2 - 2
2*a^3 - a^2 - 2
p11=2*a^3 - a^2 - 2 
       
minpoly(p11).disc().factor() 
        
3^4 * 5^3 * 11^2
3^4 * 5^3 * 11^2
minpoly(a*p11).disc().factor() 
        
5^3 * 11^2 * 19^2
5^3 * 11^2 * 19^2
minpoly(a^2*p11).disc().factor() 
        
5^3 * 11^2
5^3 * 11^2
minpoly(a^3*p11).disc().factor() 
        
2^12 * 5^3 * 11^2
2^12 * 5^3 * 11^2
minpoly(a^4*p11).disc().factor() 
        
5^9 * 11^2
5^9 * 11^2
K.factor(31) 
        
(Fractional ideal (2*a^2 - a)) * (Fractional ideal (2*a^3 - 1)) *
(Fractional ideal (-2*a^3 + a)) * (Fractional ideal (a^3 - 2*a^2))
(Fractional ideal (2*a^2 - a)) * (Fractional ideal (2*a^3 - 1)) * (Fractional ideal (-2*a^3 + a)) * (Fractional ideal (a^3 - 2*a^2))
L.factor(31) 
        
(Fractional ideal (5*t + 3)) * (Fractional ideal (5*t + 2))
(Fractional ideal (5*t + 3)) * (Fractional ideal (5*t + 2))
K.factor(5*a+5*a^4+3) 
        
(Fractional ideal (2*a^2 - a)) * (Fractional ideal (a^3 - 2*a^2))
(Fractional ideal (2*a^2 - a)) * (Fractional ideal (a^3 - 2*a^2))
p31=(2*a^2 - a)*(2*a^3 - 1) 
       
minpoly(p31).disc().factor() 
        
5^3 * 11^2 * 31^2
5^3 * 11^2 * 31^2
minpoly(a*p31).disc().factor() 
        
5^9 * 31^2
5^9 * 31^2
minpoly(a^2*p31).disc().factor() 
        
3^12 * 5^3 * 31^2
3^12 * 5^3 * 31^2
minpoly(a^3*p31).disc().factor() 
        
5^3 * 31^2 * 41^2
5^3 * 31^2 * 41^2
minpoly(a^4*p31).disc().factor() 
        
2^12 * 5^3 * 11^2 * 31^2
2^12 * 5^3 * 11^2 * 31^2
K.factor(41) 
        
(Fractional ideal (-3*a^3 - 2*a^2 - a - 1)) * (Fractional ideal (a^3 +
2*a^2 - a)) * (Fractional ideal (-2*a^3 - a^2 + a - 1)) * (Fractional
ideal (2*a^3 + 3*a^2 + a + 2))
(Fractional ideal (-3*a^3 - 2*a^2 - a - 1)) * (Fractional ideal (a^3 + 2*a^2 - a)) * (Fractional ideal (-2*a^3 - a^2 + a - 1)) * (Fractional ideal (2*a^3 + 3*a^2 + a + 2))
L.factor(41) 
        
(Fractional ideal (t - 6)) * (Fractional ideal (t + 7))
(Fractional ideal (t - 6)) * (Fractional ideal (t + 7))
K.factor(a+a^4-6) 
        
(Fractional ideal (-2*a^3 - a^2 + a - 1)) * (Fractional ideal (2*a^3 +
3*a^2 + a + 2))
(Fractional ideal (-2*a^3 - a^2 + a - 1)) * (Fractional ideal (2*a^3 + 3*a^2 + a + 2))
p41=(-2*a^3 - a^2 + a - 1)*(-3*a^3 - 2*a^2 - a - 1) 
       
minpoly(p41).disc().factor() 
        
5^3 * 31^2 * 41^2
5^3 * 31^2 * 41^2
minpoly(a*p41).disc().factor() 
        
2^16 * 5^3 * 41^2
2^16 * 5^3 * 41^2
minpoly(a^2*p41).disc().factor() 
        
5^3 * 11^4 * 41^2
5^3 * 11^4 * 41^2
minpoly(a^3*p41).disc().factor() 
        
5^3 * 41^2 * 59^2
5^3 * 41^2 * 59^2
minpoly(a^4*p41).disc().factor() 
        
3^4 * 5^9 * 41^2
3^4 * 5^9 * 41^2
a^3*p31 
        
3*a^3 - 3*a^2 - a - 3
3*a^3 - 3*a^2 - a - 3
p41 
        
3*a^3 + a^2 + 5*a + 7
3*a^3 + a^2 + 5*a + 7