Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(203\)\(\medspace = 7 \cdot 29 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.1698181681.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.203.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-7}, \sqrt{29})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 16x^{4} - 70x^{3} + 49x^{2} - 98x + 196 \)
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The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 21\cdot 53 + 4\cdot 53^{2} + 19\cdot 53^{3} + 36\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 26\cdot 53 + 16\cdot 53^{2} + 38\cdot 53^{3} + 7\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 2\cdot 53 + 41\cdot 53^{2} + 33\cdot 53^{3} + 38\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 3\cdot 53 + 44\cdot 53^{2} + 14\cdot 53^{3} + 23\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 39\cdot 53 + 20\cdot 53^{2} + 5\cdot 53^{3} + 44\cdot 53^{4} +O(53^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 35 + 44\cdot 53 + 32\cdot 53^{2} + 24\cdot 53^{3} + 15\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 44 + 36\cdot 53 + 24\cdot 53^{2} + 47\cdot 53^{3} + 30\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 50 + 37\cdot 53 + 27\cdot 53^{2} + 28\cdot 53^{3} + 15\cdot 53^{4} +O(53^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,3,2,4)(5,7,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.