Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.9834496.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.56.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{6} + x^{4} - 6x^{2} + 4 \)
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The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 9\cdot 23 + 19\cdot 23^{2} + 11\cdot 23^{3} + 5\cdot 23^{4} +O(23^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 7 + 6\cdot 23 + 7\cdot 23^{2} + 2\cdot 23^{3} + 18\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 + 16\cdot 23 + 2\cdot 23^{2} + 18\cdot 23^{3} + 14\cdot 23^{4} +O(23^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 10 + 15\cdot 23 + 5\cdot 23^{3} + 12\cdot 23^{4} +O(23^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 13 + 7\cdot 23 + 22\cdot 23^{2} + 17\cdot 23^{3} + 10\cdot 23^{4} +O(23^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 15 + 6\cdot 23 + 20\cdot 23^{2} + 4\cdot 23^{3} + 8\cdot 23^{4} +O(23^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 16 + 16\cdot 23 + 15\cdot 23^{2} + 20\cdot 23^{3} + 4\cdot 23^{4} +O(23^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 17 + 13\cdot 23 + 3\cdot 23^{2} + 11\cdot 23^{3} + 17\cdot 23^{4} +O(23^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,5,7)(2,8,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.