Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(407\)\(\medspace = 11 \cdot 37 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.741610573.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.407.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.4477.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - x^{6} - x^{5} + 9x^{4} - 15x^{3} + 20x^{2} - 8x + 1 \)
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The roots of $f$ are computed in $\Q_{ 433 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 285\cdot 433 + 212\cdot 433^{2} + 146\cdot 433^{3} + 413\cdot 433^{4} +O(433^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 184\cdot 433 + 49\cdot 433^{2} + 247\cdot 433^{3} + 80\cdot 433^{4} +O(433^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 145 + 43\cdot 433 + 141\cdot 433^{2} + 406\cdot 433^{3} + 226\cdot 433^{4} +O(433^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 267 + 419\cdot 433 + 191\cdot 433^{2} + 7\cdot 433^{3} + 420\cdot 433^{4} +O(433^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 287 + 377\cdot 433 + 96\cdot 433^{2} + 235\cdot 433^{3} + 209\cdot 433^{4} +O(433^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 314 + 418\cdot 433 + 64\cdot 433^{2} + 311\cdot 433^{3} + 413\cdot 433^{4} +O(433^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 338 + 98\cdot 433 + 209\cdot 433^{2} + 116\cdot 433^{3} + 257\cdot 433^{4} +O(433^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 344 + 337\cdot 433 + 332\cdot 433^{2} + 261\cdot 433^{3} + 143\cdot 433^{4} +O(433^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(3,6)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $2$ | $8$ | $(1,6,4,2,8,3,5,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,2,5,6,8,7,4,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.