Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1764\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.38423222208.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.0.12348.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 43 + 102\cdot 137 + 36\cdot 137^{2} + 134\cdot 137^{3} + 60\cdot 137^{4} + 32\cdot 137^{5} +O(137^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 47 + 4\cdot 137 + 130\cdot 137^{2} + 89\cdot 137^{3} + 97\cdot 137^{4} + 106\cdot 137^{5} +O(137^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 49 + 60\cdot 137 + 100\cdot 137^{2} + 12\cdot 137^{3} + 124\cdot 137^{4} + 98\cdot 137^{5} +O(137^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 56 + 73\cdot 137 + 112\cdot 137^{2} + 123\cdot 137^{3} + 33\cdot 137^{4} + 81\cdot 137^{5} +O(137^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 74 + 105\cdot 137 + 114\cdot 137^{2} + 116\cdot 137^{3} + 135\cdot 137^{4} + 20\cdot 137^{5} +O(137^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 86 + 100\cdot 137 + 43\cdot 137^{2} + 26\cdot 137^{3} + 116\cdot 137^{4} + 19\cdot 137^{5} +O(137^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 87 + 24\cdot 137 + 67\cdot 137^{2} + 40\cdot 137^{3} + 76\cdot 137^{4} + 93\cdot 137^{5} +O(137^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 109 + 76\cdot 137 + 79\cdot 137^{2} + 3\cdot 137^{3} + 40\cdot 137^{4} + 94\cdot 137^{5} +O(137^{6})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)(7,8)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,6)(3,5)(4,8)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,6)(4,5)(7,8)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,6,4,5)(2,7,3,8)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,8,5,3,4,7,6,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,6,8,4,2,5,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |