Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(441\)\(\medspace = 3^{2} \cdot 7^{2} \) |
| Artin stem field: | Galois closure of 8.2.257298363.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.189.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + x^{6} - 5x^{5} + 7x^{4} - 2x^{3} + x^{2} - 5x + 1 \)
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The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 39\cdot 79 + 56\cdot 79^{2} + 40\cdot 79^{3} + 4\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 22\cdot 79 + 60\cdot 79^{2} + 46\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 13 + 13\cdot 79 + 3\cdot 79^{2} + 65\cdot 79^{3} + 28\cdot 79^{4} +O(79^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 14 + 25\cdot 79 + 49\cdot 79^{2} + 17\cdot 79^{3} +O(79^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 39 + 53\cdot 79 + 46\cdot 79^{2} + 48\cdot 79^{3} + 12\cdot 79^{4} +O(79^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 49 + 71\cdot 79 + 70\cdot 79^{2} + 52\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 51 + 11\cdot 79 + 21\cdot 79^{2} + 34\cdot 79^{3} + 44\cdot 79^{4} +O(79^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 56 + 8\cdot 79^{2} + 10\cdot 79^{3} + 72\cdot 79^{4} +O(79^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(3,8)(4,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,4,2,6)(3,8,5,7)$ | $0$ |
| $4$ | $4$ | $(1,8,2,7)(3,6,5,4)$ | $0$ |
| $2$ | $8$ | $(1,3,4,8,2,5,6,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,4,7,2,3,6,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.