Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.796594176.14 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.84.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{7})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 4x^{6} + 12x^{5} - 21x^{4} + 36x^{3} + 36x^{2} - 54x + 81 \)
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The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 24\cdot 31 + 22\cdot 31^{2} + 24\cdot 31^{3} + 5\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 21\cdot 31 + 16\cdot 31^{2} + 24\cdot 31^{3} + 10\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 + 10\cdot 31 + 9\cdot 31^{2} + 4\cdot 31^{3} + 16\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 + 17\cdot 31 + 31^{2} + 4\cdot 31^{3} + 10\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 + 17\cdot 31 + 28\cdot 31^{2} + 18\cdot 31^{3} + 22\cdot 31^{4} +O(31^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 14 + 26\cdot 31 + 2\cdot 31^{3} + 11\cdot 31^{4} +O(31^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 21 + 31 + 22\cdot 31^{2} + 17\cdot 31^{3} + 29\cdot 31^{4} +O(31^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 29 + 5\cdot 31 + 22\cdot 31^{2} + 27\cdot 31^{3} + 17\cdot 31^{4} +O(31^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,7,5,6)(2,4,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.