Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(712\)\(\medspace = 2^{3} \cdot 89 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.2887553024.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.712.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.5696.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} + 13x^{4} - 24x^{3} + 20x^{2} - 12x + 9 \)
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The roots of $f$ are computed in $\Q_{ 179 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 30 + 6\cdot 179 + 42\cdot 179^{2} + 96\cdot 179^{3} + 64\cdot 179^{4} + 63\cdot 179^{5} +O(179^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 31 + 61\cdot 179 + 138\cdot 179^{2} + 101\cdot 179^{3} + 101\cdot 179^{4} + 121\cdot 179^{5} +O(179^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 113 + 27\cdot 179 + 17\cdot 179^{2} + 128\cdot 179^{3} + 62\cdot 179^{4} + 131\cdot 179^{5} +O(179^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 116 + 46\cdot 179 + 4\cdot 179^{2} + 111\cdot 179^{3} + 99\cdot 179^{4} + 31\cdot 179^{5} +O(179^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 126 + 96\cdot 179 + 49\cdot 179^{2} + 159\cdot 179^{3} + 171\cdot 179^{4} + 113\cdot 179^{5} +O(179^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 156 + 38\cdot 179 + 147\cdot 179^{2} + 32\cdot 179^{3} + 32\cdot 179^{4} + 66\cdot 179^{5} +O(179^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 159 + 103\cdot 179 + 145\cdot 179^{2} + 111\cdot 179^{3} + 177\cdot 179^{4} + 133\cdot 179^{5} +O(179^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 164 + 155\cdot 179 + 171\cdot 179^{2} + 153\cdot 179^{3} + 5\cdot 179^{4} + 54\cdot 179^{5} +O(179^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $4$ | $2$ | $(1,8)(4,5)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,5,4,8)(2,6,3,7)$ | $0$ |
| $2$ | $8$ | $(1,7,5,2,4,6,8,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,2,8,7,4,3,5,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.