Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
| Artin stem field: | Galois closure of 8.0.8719893.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.273.4t1.a.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.322959.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} - x^{6} + 11x^{5} - 6x^{4} - 10x^{3} + 9x^{2} - x + 1 \)
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The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 22\cdot 181 + 119\cdot 181^{2} + 127\cdot 181^{3} + 146\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 109\cdot 181 + 174\cdot 181^{2} + 179\cdot 181^{3} + 64\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 + 155\cdot 181 + 70\cdot 181^{2} + 151\cdot 181^{3} + 124\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 + 145\cdot 181 + 181^{2} + 36\cdot 181^{3} + 118\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 67 + 89\cdot 181 + 109\cdot 181^{2} + 58\cdot 181^{3} + 121\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 119 + 103\cdot 181 + 47\cdot 181^{2} + 136\cdot 181^{3} + 85\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 127 + 82\cdot 181 + 52\cdot 181^{2} + 106\cdot 181^{3} + 157\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 142 + 16\cdot 181 + 148\cdot 181^{2} + 108\cdot 181^{3} + 85\cdot 181^{4} +O(181^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,5)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,6,5,4)(2,8,7,3)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,5,6)(2,3,7,8)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,5,4)(2,3,7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,5,6)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,5,4)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,5,4)(2,7)(3,8)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,4,5,6)(2,7)(3,8)$ | $-\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,7,5,2)(3,4,8,6)$ | $0$ |
| $4$ | $8$ | $(1,7,6,3,5,2,4,8)$ | $0$ |
| $4$ | $8$ | $(1,3,4,7,5,8,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.