Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(376\)\(\medspace = 2^{3} \cdot 47 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.425259008.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.376.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.3008.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - x^{6} + 2x^{5} + 6x^{4} - 14x^{3} + 9x^{2} - 6x + 1 \)
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The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 46\cdot 103 + 47\cdot 103^{2} + 91\cdot 103^{3} + 6\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 52\cdot 103 + 21\cdot 103^{2} + 29\cdot 103^{3} + 99\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 34 + 30\cdot 103 + 64\cdot 103^{2} + 85\cdot 103^{3} + 56\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 38 + 41\cdot 103 + 41\cdot 103^{2} + 52\cdot 103^{3} + 66\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 64 + 32\cdot 103 + 14\cdot 103^{2} + 65\cdot 103^{3} + 29\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 73 + 9\cdot 103 + 29\cdot 103^{2} + 91\cdot 103^{3} + 55\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 83 + 8\cdot 103 + 77\cdot 103^{2} + 23\cdot 103^{3} + 12\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 97 + 87\cdot 103 + 13\cdot 103^{2} + 76\cdot 103^{3} + 84\cdot 103^{4} +O(103^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,3)(4,7)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,3)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,6)(2,7,3,4)$ | $0$ |
| $2$ | $8$ | $(1,7,6,2,8,4,5,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,2,5,7,8,3,6,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.