Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(583\)\(\medspace = 11 \cdot 53 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.2179708157.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.583.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.6413.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 5x^{6} - x^{5} + 15x^{4} + 4x^{3} + 25x^{2} + 22x + 5 \)
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The roots of $f$ are computed in $\Q_{ 587 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 120 + 509\cdot 587 + 70\cdot 587^{2} + 134\cdot 587^{3} + 206\cdot 587^{4} +O(587^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 255 + 420\cdot 587 + 201\cdot 587^{2} + 335\cdot 587^{3} + 484\cdot 587^{4} +O(587^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 308 + 193\cdot 587 + 206\cdot 587^{2} + 299\cdot 587^{3} + 134\cdot 587^{4} +O(587^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 322 + 444\cdot 587 + 282\cdot 587^{2} + 67\cdot 587^{3} + 64\cdot 587^{4} +O(587^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 340 + 250\cdot 587 + 519\cdot 587^{2} + 32\cdot 587^{3} + 235\cdot 587^{4} +O(587^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 486 + 364\cdot 587 + 257\cdot 587^{2} + 381\cdot 587^{3} + 104\cdot 587^{4} +O(587^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 548 + 31\cdot 587 + 515\cdot 587^{2} + 252\cdot 587^{3} + 6\cdot 587^{4} +O(587^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 557 + 132\cdot 587 + 294\cdot 587^{2} + 257\cdot 587^{3} + 525\cdot 587^{4} +O(587^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,8,7,2)(3,4,5,6)$ | $0$ |
| $2$ | $8$ | $(1,3,2,6,7,5,8,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,6,8,3,7,4,2,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.