Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(208\)\(\medspace = 2^{4} \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.116985856.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.52.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{13})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 8x^{6} - 6x^{5} + 7x^{4} - 10x^{3} + 2x^{2} - 6x + 9 \)
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The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 53 + 39\cdot 53^{2} + 11\cdot 53^{3} + 5\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 35\cdot 53 + 45\cdot 53^{2} + 25\cdot 53^{3} + 8\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 17\cdot 53 + 9\cdot 53^{2} + 7\cdot 53^{3} + 5\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 8\cdot 53 + 4\cdot 53^{2} + 37\cdot 53^{3} + 50\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 + 26\cdot 53 + 50\cdot 53^{3} + 44\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 23 + 8\cdot 53 + 17\cdot 53^{2} + 31\cdot 53^{3} + 41\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 25 + 32\cdot 53 + 14\cdot 53^{2} + 39\cdot 53^{3} + 13\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 52 + 29\cdot 53 + 28\cdot 53^{2} + 9\cdot 53^{3} + 42\cdot 53^{4} +O(53^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,2,4,6)(3,8,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.