Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1216\)\(\medspace = 2^{6} \cdot 19 \) |
| Artin stem field: | Galois closure of 8.0.378535936.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.152.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{-19})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 20x^{6} - 44x^{5} + 98x^{4} - 128x^{3} + 144x^{2} - 96x + 34 \)
|
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 30 + 54\cdot 137 + 7\cdot 137^{2} + 122\cdot 137^{3} + 33\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 55 + 86\cdot 137 + 97\cdot 137^{2} + 55\cdot 137^{3} + 35\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 71 + 34\cdot 137 + 42\cdot 137^{2} + 87\cdot 137^{3} + 72\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 86 + 3\cdot 137 + 105\cdot 137^{2} + 126\cdot 137^{3} + 73\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 89 + 44\cdot 137 + 119\cdot 137^{2} + 74\cdot 137^{3} + 93\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 103 + 27\cdot 137 + 87\cdot 137^{2} + 77\cdot 137^{3} + 114\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 120 + 98\cdot 137 + 126\cdot 137^{2} + 8\cdot 137^{3} + 132\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 135 + 60\cdot 137 + 99\cdot 137^{2} + 131\cdot 137^{3} + 128\cdot 137^{4} +O(137^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,5,3,4)(2,6,7,8)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,3,5)(2,8,7,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,3,2)(4,6,5,8)$ | $0$ |
| $2$ | $4$ | $(1,6,3,8)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,4,3,5)(2,6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.