Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1104\)\(\medspace = 2^{4} \cdot 3 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.4036718592.6 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.276.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.3312.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 4x^{6} + 4x^{5} - 18x^{4} + 32x^{3} - 32x^{2} + 8x + 4 \)
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The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 8\cdot 71 + 38\cdot 71^{2} + 69\cdot 71^{3} + 21\cdot 71^{4} + 22\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 27\cdot 71 + 44\cdot 71^{2} + 27\cdot 71^{3} + 8\cdot 71^{4} + 28\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 28 + 62\cdot 71 + 7\cdot 71^{2} + 23\cdot 71^{3} + 9\cdot 71^{4} + 61\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 + 33\cdot 71 + 51\cdot 71^{3} + 17\cdot 71^{4} + 55\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 36 + 19\cdot 71 + 3\cdot 71^{2} + 40\cdot 71^{3} + 60\cdot 71^{4} + 59\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 49 + 52\cdot 71 + 6\cdot 71^{2} + 2\cdot 71^{3} + 29\cdot 71^{4} + 60\cdot 71^{5} +O(71^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 53 + 32\cdot 71 + 23\cdot 71^{2} + 9\cdot 71^{3} + 7\cdot 71^{4} + 12\cdot 71^{5} +O(71^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 54 + 47\cdot 71 + 17\cdot 71^{2} + 61\cdot 71^{3} + 58\cdot 71^{4} + 55\cdot 71^{5} +O(71^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,6)(3,5)(4,8)$ | $0$ |
| $4$ | $2$ | $(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,2,3,8)(4,5,6,7)$ | $0$ |
| $2$ | $8$ | $(1,6,8,5,3,4,2,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,2,6,3,7,8,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.