Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(196\)\(\medspace = 2^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.30118144.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{7})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 2x^{6} + 4x^{5} - x^{4} + 4x^{3} + 2x^{2} - 2x + 1 \)
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The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 46\cdot 53 + 29\cdot 53^{2} + 27\cdot 53^{3} + 47\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 51\cdot 53 + 27\cdot 53^{2} + 11\cdot 53^{3} + 48\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 50\cdot 53 + 14\cdot 53^{2} + 32\cdot 53^{3} + 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 19\cdot 53 + 44\cdot 53^{3} + 51\cdot 53^{4} +O(53^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 41 + 15\cdot 53 + 14\cdot 53^{2} + 52\cdot 53^{3} + 40\cdot 53^{4} +O(53^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 42 + 42\cdot 53 + 7\cdot 53^{2} + 2\cdot 53^{3} + 5\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 46 + 10\cdot 53 + 33\cdot 53^{2} + 34\cdot 53^{3} + 8\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 51 + 27\cdot 53 + 30\cdot 53^{2} + 7\cdot 53^{3} + 8\cdot 53^{4} +O(53^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,6)(3,7,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.