Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1520\)\(\medspace = 2^{4} \cdot 5 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1097440000.5 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.95.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.475.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 5x^{6} + 2x^{4} - 20x^{2} - 19 \)
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The roots of $f$ are computed in $\Q_{ 191 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 22\cdot 191 + 117\cdot 191^{2} + 125\cdot 191^{3} + 136\cdot 191^{4} + 83\cdot 191^{5} +O(191^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 30 + 40\cdot 191 + 171\cdot 191^{2} + 106\cdot 191^{3} + 116\cdot 191^{4} + 158\cdot 191^{5} +O(191^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 37 + 26\cdot 191 + 11\cdot 191^{2} + 96\cdot 191^{3} + 144\cdot 191^{4} + 104\cdot 191^{5} +O(191^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 68 + 114\cdot 191 + 45\cdot 191^{2} + 159\cdot 191^{3} + 13\cdot 191^{4} + 77\cdot 191^{5} +O(191^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 123 + 76\cdot 191 + 145\cdot 191^{2} + 31\cdot 191^{3} + 177\cdot 191^{4} + 113\cdot 191^{5} +O(191^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 154 + 164\cdot 191 + 179\cdot 191^{2} + 94\cdot 191^{3} + 46\cdot 191^{4} + 86\cdot 191^{5} +O(191^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 161 + 150\cdot 191 + 19\cdot 191^{2} + 84\cdot 191^{3} + 74\cdot 191^{4} + 32\cdot 191^{5} +O(191^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 178 + 168\cdot 191 + 73\cdot 191^{2} + 65\cdot 191^{3} + 54\cdot 191^{4} + 107\cdot 191^{5} +O(191^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,6,7,4,8,3,2,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,2,6,8,5,7,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.