Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(539\)\(\medspace = 7^{2} \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1096135733.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.11.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.3773.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 7x^{2} - 2x + 4 \)
|
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 117\cdot 137 + 114\cdot 137^{2} + 91\cdot 137^{3} + 39\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 60 + 23\cdot 137 + 86\cdot 137^{2} + 129\cdot 137^{3} + 112\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 61 + 110\cdot 137 + 95\cdot 137^{2} + 86\cdot 137^{3} + 7\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 79 + 63\cdot 137 + 115\cdot 137^{2} + 25\cdot 137^{3} + 18\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 98 + 126\cdot 137 + 49\cdot 137^{2} + 135\cdot 137^{3} + 131\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 119 + 3\cdot 137 + 40\cdot 137^{2} + 111\cdot 137^{3} + 40\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 130 + 119\cdot 137 + 84\cdot 137^{2} + 69\cdot 137^{3} + 71\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 135 + 119\cdot 137 + 97\cdot 137^{2} + 34\cdot 137^{3} + 125\cdot 137^{4} +O(137^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,7)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,6)(2,3)(4,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,3)(4,7)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,3,4,7)(2,6,8,5)$ | $0$ |
| $2$ | $8$ | $(1,5,7,8,4,6,3,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,8,3,5,4,2,7,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.