Properties

Label 2.120.8t11.c.b
Dimension 22
Group Q8:C2Q_8:C_2
Conductor 120120
Root number not computed
Indicator 00

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Basic invariants

Dimension: 22
Group: Q8:C2Q_8:C_2
Conductor: 120120=2335\medspace = 2^{3} \cdot 3 \cdot 5
Artin stem field: Galois closure of 8.0.3240000.1
Galois orbit size: 22
Smallest permutation container: Q8:C2Q_8:C_2
Parity: odd
Determinant: 1.120.2t1.b.a
Projective image: C22C_2^2
Projective field: Galois closure of Q(6,10)\Q(\sqrt{-6}, \sqrt{10})

Defining polynomial

f(x)f(x)== x83x7+3x6+x52x43x3+7x24x+1 x^{8} - 3x^{7} + 3x^{6} + x^{5} - 2x^{4} - 3x^{3} + 7x^{2} - 4x + 1 Copy content Toggle raw display .

The roots of ff are computed in Q79\Q_{ 79 } to precision 5.

Roots:
r1r_{ 1 } == 7+1979+44792+40793+47794+O(795) 7 + 19\cdot 79 + 44\cdot 79^{2} + 40\cdot 79^{3} + 47\cdot 79^{4} +O(79^{5}) Copy content Toggle raw display
r2r_{ 2 } == 17+7479+66792+61793+63794+O(795) 17 + 74\cdot 79 + 66\cdot 79^{2} + 61\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5}) Copy content Toggle raw display
r3r_{ 3 } == 19+7879+12792+15793+56794+O(795) 19 + 78\cdot 79 + 12\cdot 79^{2} + 15\cdot 79^{3} + 56\cdot 79^{4} +O(79^{5}) Copy content Toggle raw display
r4r_{ 4 } == 32+6679+21792+70793+68794+O(795) 32 + 66\cdot 79 + 21\cdot 79^{2} + 70\cdot 79^{3} + 68\cdot 79^{4} +O(79^{5}) Copy content Toggle raw display
r5r_{ 5 } == 33+4479+78792+65793+37794+O(795) 33 + 44\cdot 79 + 78\cdot 79^{2} + 65\cdot 79^{3} + 37\cdot 79^{4} +O(79^{5}) Copy content Toggle raw display
r6r_{ 6 } == 38+7279+68792+13793+12794+O(795) 38 + 72\cdot 79 + 68\cdot 79^{2} + 13\cdot 79^{3} + 12\cdot 79^{4} +O(79^{5}) Copy content Toggle raw display
r7r_{ 7 } == 40+4579+63792+67793+17794+O(795) 40 + 45\cdot 79 + 63\cdot 79^{2} + 67\cdot 79^{3} + 17\cdot 79^{4} +O(79^{5}) Copy content Toggle raw display
r8r_{ 8 } == 54+7379+37792+59793+11794+O(795) 54 + 73\cdot 79 + 37\cdot 79^{2} + 59\cdot 79^{3} + 11\cdot 79^{4} +O(79^{5}) Copy content Toggle raw display

Generators of the action on the roots r1,,r8r_1, \ldots, r_{ 8 }

Cycle notation
(1,6)(2,8)(3,7)(4,5)(1,6)(2,8)(3,7)(4,5)
(3,7)(4,5)(3,7)(4,5)
(1,4,6,5)(2,7,8,3)(1,4,6,5)(2,7,8,3)
(1,3,6,7)(2,4,8,5)(1,3,6,7)(2,4,8,5)

Character values on conjugacy classes

SizeOrderAction on r1,,r8r_1, \ldots, r_{ 8 } Character valueComplex conjugation
1111()()22
1122(1,6)(2,8)(3,7)(4,5)(1,6)(2,8)(3,7)(4,5)2-2
2222(3,7)(4,5)(3,7)(4,5)00
2222(1,4)(2,7)(3,8)(5,6)(1,4)(2,7)(3,8)(5,6)00
2222(1,3)(2,4)(5,8)(6,7)(1,3)(2,4)(5,8)(6,7)00
1144(1,8,6,2)(3,5,7,4)(1,8,6,2)(3,5,7,4)2ζ42 \zeta_{4}
1144(1,2,6,8)(3,4,7,5)(1,2,6,8)(3,4,7,5)2ζ4-2 \zeta_{4}
2244(1,4,6,5)(2,7,8,3)(1,4,6,5)(2,7,8,3)00
2244(1,3,6,7)(2,4,8,5)(1,3,6,7)(2,4,8,5)00
2244(1,8,6,2)(3,4,7,5)(1,8,6,2)(3,4,7,5)00