1745.sagews

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

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))

(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))

K.factor(7*a+7*a^4+4)

(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

minpoly(p1).disc().factor()

5^3 * 61^2

minpoly(a*p1).disc().factor()

5^9 * 19^2 * 61^2

minpoly(a^2*p1).disc().factor()

2^20 * 5^3 * 61^2

minpoly(a^3*p1).disc().factor()

5^3 * 11^4 * 41^2 * 61^2

minpoly(a^4*p1).disc().factor()

3^12 * 5^3 * 61^2

minpoly(b1).disc().factor()

5^7 * 11^2

minpoly(a*b1).disc().factor()

3^12 * 5^3

minpoly(a^2*b1).disc().factor()

2^4 * 5^3 * 11^4

minpoly(a^3*b1).disc().factor()

5^3

minpoly(a^4*b1).disc().factor()

5^3 * 11^2 * 31^2

minpoly(b2).disc()

14535125

minpoly(a*b2).disc().factor()

3^12 * 5^3

minpoly(a^2*b2).disc().factor()

5^3

minpoly(b3).disc().factor()

2^4 * 5^3 * 11^4

minpoly(b4).disc().factor()

3^12 * 5^3

a^3*b1

a^2 + 3

minpoly(a^3*b1)

x^4 - 11*x^3 + 46*x^2 - 86*x + 61

minpoly(p1)

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

minpoly(1+p1)

x^4 + 25*x^3 + 250*x^2 + 1190*x + 2255

is_prime(5851)

True

a^3*p1

-8*a^3 - 2*a^2 + a - 2

a^4*b1

a^3 + 3*a

is_prime(311)

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))

L.factor(311)

(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))

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

minpoly(p2^5).disc().factor()

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

minpoly(a^2*p2).disc().factor()

5^3 * 19^4 * 109^2 * 331^2

minpoly(a^3*p2).disc().factor()

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

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

Mod(17,29)^4

1

3

Mod(3,31)^5

26

for i in [1..5]:
 print Mod(26,31)^i

26 25 30 5 6

Mod(2*8*12*6*14*4,31)^2

1

is_prime(1861)

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))

L.factor(1861)

(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))

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

Mod(221096161914265125,31)

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

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))

L.factor(2791)

(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))

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

Mod(minpoly(p2791^5).disc(),31)

20

Mod(20*21*19*20,31)^2

20

Mod(20*25*19*24*5,31)^2

5

Mod(21*25*20*24*4,31)^2

16

Mod(20*21*25*5*4,31)^2

5

Mod(19*20*24*5*4,31)^2

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

Mod(20,31)^5

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))

L.factor(11)

(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))

(-2*a^3 - 2*a^2 - a - 1)*(-a^3 - 2*a - 1)

2*a^3 - a^2 - 2

p11=2*a^3 - a^2 - 2

minpoly(p11).disc().factor()

3^4 * 5^3 * 11^2

minpoly(a*p11).disc().factor()

5^3 * 11^2 * 19^2

minpoly(a^2*p11).disc().factor()

5^3 * 11^2

minpoly(a^3*p11).disc().factor()

2^12 * 5^3 * 11^2

minpoly(a^4*p11).disc().factor()

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))

L.factor(31)

(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))

p31=(2*a^2 - a)*(2*a^3 - 1)

minpoly(p31).disc().factor()

5^3 * 11^2 * 31^2

minpoly(a*p31).disc().factor()

5^9 * 31^2

minpoly(a^2*p31).disc().factor()

3^12 * 5^3 * 31^2

minpoly(a^3*p31).disc().factor()

5^3 * 31^2 * 41^2

minpoly(a^4*p31).disc().factor()

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))

L.factor(41)

(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))

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

minpoly(a*p41).disc().factor()

2^16 * 5^3 * 41^2

minpoly(a^2*p41).disc().factor()

5^3 * 11^4 * 41^2

minpoly(a^3*p41).disc().factor()

5^3 * 41^2 * 59^2

minpoly(a^4*p41).disc().factor()

3^4 * 5^9 * 41^2

a^3*p31

3*a^3 - 3*a^2 - a - 3

p41

3*a^3 + a^2 + 5*a + 7