Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1155\)\(\medspace = 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Artin stem field: | Galois closure of 8.0.1634180625.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.1155.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-35}, \sqrt{165})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 2x^{6} + 6x^{5} + 4x^{4} - 18x^{3} + 17x^{2} - 6x + 4 \)
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The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 59 + 132\cdot 149 + 18\cdot 149^{2} + 50\cdot 149^{3} + 138\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 77 + 78\cdot 149 + 47\cdot 149^{2} + 59\cdot 149^{3} + 18\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 113 + 11\cdot 149 + 132\cdot 149^{2} + 84\cdot 149^{3} + 56\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 115 + 138\cdot 149 + 141\cdot 149^{2} + 17\cdot 149^{3} + 89\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 122 + 131\cdot 149 + 147\cdot 149^{2} + 110\cdot 149^{3} + 61\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 131 + 106\cdot 149 + 11\cdot 149^{2} + 94\cdot 149^{3} + 85\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 136 + 75\cdot 149 + 119\cdot 149^{2} + 42\cdot 149^{3} + 12\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 143 + 68\cdot 149 + 125\cdot 149^{2} + 135\cdot 149^{3} + 133\cdot 149^{4} +O(149^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,3)(4,8)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,6)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,4,6,8)(2,5,3,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,6,4)(2,7,3,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,6,2)(4,7,8,5)$ | $0$ |
| $2$ | $4$ | $(1,5,6,7)(2,4,3,8)$ | $0$ |
| $2$ | $4$ | $(1,8,6,4)(2,5,3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.