Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(452\)\(\medspace = 2^{2} \cdot 113 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.369381632.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.452.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.1808.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 7x^{6} - 10x^{5} + 3x^{2} + 4x + 1 \)
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The roots of $f$ are computed in $\Q_{ 149 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 34\cdot 149 + 145\cdot 149^{2} + 97\cdot 149^{3} + 142\cdot 149^{4} + 82\cdot 149^{5} +O(149^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 7 + 64\cdot 149 + 14\cdot 149^{2} + 18\cdot 149^{3} + 111\cdot 149^{4} + 131\cdot 149^{5} +O(149^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 15 + 56\cdot 149 + 133\cdot 149^{2} + 127\cdot 149^{3} + 56\cdot 149^{4} + 50\cdot 149^{5} +O(149^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 17 + 115\cdot 149 + 57\cdot 149^{2} + 51\cdot 149^{3} + 7\cdot 149^{4} + 79\cdot 149^{5} +O(149^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 22 + 67\cdot 149 + 111\cdot 149^{2} + 38\cdot 149^{3} + 69\cdot 149^{4} + 69\cdot 149^{5} +O(149^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 50 + 39\cdot 149 + 11\cdot 149^{2} + 136\cdot 149^{3} + 95\cdot 149^{4} + 66\cdot 149^{5} +O(149^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 86 + 41\cdot 149 + 45\cdot 149^{2} + 99\cdot 149^{3} + 48\cdot 149^{4} + 79\cdot 149^{5} +O(149^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 101 + 29\cdot 149 + 77\cdot 149^{2} + 26\cdot 149^{3} + 64\cdot 149^{4} + 36\cdot 149^{5} +O(149^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(4,6)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,5,8)(2,4,3,6)$ | $0$ |
| $2$ | $8$ | $(1,6,7,2,5,4,8,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,2,8,6,5,3,7,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.