Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(475\)\(\medspace = 5^{2} \cdot 19 \) |
| Artin stem field: | Galois closure of 8.2.107171875.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.475.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 2x^{6} + x^{4} + 5x^{3} - 7x^{2} - 6x + 1 \)
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The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 86\cdot 199 + 183\cdot 199^{2} + 102\cdot 199^{3} + 51\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 + 64\cdot 199 + 101\cdot 199^{2} + 44\cdot 199^{3} + 158\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 42 + 179\cdot 199 + 134\cdot 199^{2} + 41\cdot 199^{3} + 53\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 83 + 38\cdot 199 + 130\cdot 199^{2} + 97\cdot 199^{3} + 9\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 149 + 176\cdot 199 + 136\cdot 199^{2} + 11\cdot 199^{3} + 129\cdot 199^{4} +O(199^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 156 + 18\cdot 199 + 141\cdot 199^{2} + 103\cdot 199^{3} + 124\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 164 + 96\cdot 199 + 146\cdot 199^{2} + 185\cdot 199^{3} + 8\cdot 199^{4} +O(199^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 171 + 135\cdot 199 + 20\cdot 199^{2} + 9\cdot 199^{3} + 62\cdot 199^{4} +O(199^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,3)(4,5)(6,8)$ | $-2$ |
| $4$ | $2$ | $(2,8)(3,6)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,5,7,4)(2,6,3,8)$ | $0$ |
| $4$ | $4$ | $(1,2,7,3)(4,6,5,8)$ | $0$ |
| $2$ | $8$ | $(1,8,5,2,7,6,4,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,6,5,3,7,8,4,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.