Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.342102016.4 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.136.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{17})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 50x^{4} - 74x^{3} + 59x^{2} - 30x + 9 \)
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The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 26\cdot 43 + 40\cdot 43^{2} + 11\cdot 43^{3} + 26\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 22\cdot 43 + 9\cdot 43^{2} + 3\cdot 43^{3} + 40\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 5\cdot 43 + 6\cdot 43^{2} + 38\cdot 43^{3} + 40\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 10\cdot 43 + 12\cdot 43^{2} + 15\cdot 43^{3} + 26\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 33\cdot 43 + 26\cdot 43^{2} + 33\cdot 43^{3} + 32\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 32 + 26\cdot 43 + 23\cdot 43^{2} + 12\cdot 43^{3} + 36\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 37 + 37\cdot 43 + 3\cdot 43^{2} + 6\cdot 43^{3} + 19\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 38 + 8\cdot 43 + 6\cdot 43^{2} + 8\cdot 43^{3} + 36\cdot 43^{4} +O(43^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,4,6)(3,5,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.