Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.2560000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.20.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 2x^{6} + 2x^{5} - 2x^{4} + 2x^{3} + 2x^{2} - 2x + 1 \)
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The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 28\cdot 29 + 3\cdot 29^{2} + 5\cdot 29^{3} + 12\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 6\cdot 29 + 3\cdot 29^{2} + 10\cdot 29^{3} + 12\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 25\cdot 29 + 24\cdot 29^{2} + 4\cdot 29^{3} + 28\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 24\cdot 29 + 13\cdot 29^{2} + 17\cdot 29^{3} + 3\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 + 15\cdot 29 + 25\cdot 29^{2} + 23\cdot 29^{3} + 14\cdot 29^{4} +O(29^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 19 + 6\cdot 29 + 13\cdot 29^{2} + 17\cdot 29^{3} +O(29^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 25 + 11\cdot 29 + 16\cdot 29^{2} + 2\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 26 + 26\cdot 29 + 14\cdot 29^{2} + 5\cdot 29^{3} + 5\cdot 29^{4} +O(29^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,3,6,4)(2,7,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.