Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(584\)\(\medspace = 2^{3} \cdot 73 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1593413632.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.584.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.4672.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 8x^{5} - x^{4} + 4x^{3} + 8x^{2} + 8x + 4 \)
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The roots of $f$ are computed in $\Q_{ 227 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 + 184\cdot 227 + 117\cdot 227^{2} + 136\cdot 227^{3} + 72\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 68 + 86\cdot 227 + 42\cdot 227^{2} + 224\cdot 227^{3} + 136\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 77 + 15\cdot 227 + 206\cdot 227^{2} + 25\cdot 227^{3} + 113\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 86 + 47\cdot 227 + 5\cdot 227^{2} + 196\cdot 227^{3} + 107\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 131 + 193\cdot 227 + 98\cdot 227^{2} + 173\cdot 227^{3} + 70\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 145 + 219\cdot 227 + 143\cdot 227^{2} + 101\cdot 227^{3} + 202\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 163 + 211\cdot 227 + 154\cdot 227^{2} + 109\cdot 227^{3} + 111\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 216 + 176\cdot 227 + 138\cdot 227^{2} + 167\cdot 227^{3} + 92\cdot 227^{4} +O(227^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,3)(4,6)(5,7)$ | $-2$ |
| $4$ | $2$ | $(2,5)(3,7)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,6,8,4)(2,7,3,5)$ | $0$ |
| $2$ | $8$ | $(1,3,6,5,8,2,4,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,4,3,8,7,6,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.