Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(14080\)\(\medspace = 2^{8} \cdot 5 \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.2.3489136640000.8 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.17600.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 33\cdot 71^{2} + 7\cdot 71^{3} + 54\cdot 71^{4} + 41\cdot 71^{5} + 2\cdot 71^{6} + 12\cdot 71^{7} + 32\cdot 71^{8} +O(71^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 9\cdot 71 + 16\cdot 71^{2} + 38\cdot 71^{3} + 66\cdot 71^{4} + 29\cdot 71^{5} + 41\cdot 71^{6} + 55\cdot 71^{7} + 61\cdot 71^{8} +O(71^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 18\cdot 71 + 65\cdot 71^{2} + 52\cdot 71^{3} + 3\cdot 71^{4} + 55\cdot 71^{5} + 13\cdot 71^{6} + 42\cdot 71^{7} + 29\cdot 71^{8} +O(71^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 26\cdot 71 + 32\cdot 71^{2} + 6\cdot 71^{3} + 19\cdot 71^{4} + 71^{6} + 16\cdot 71^{7} + 59\cdot 71^{8} +O(71^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 56 + 44\cdot 71 + 38\cdot 71^{2} + 64\cdot 71^{3} + 51\cdot 71^{4} + 70\cdot 71^{5} + 69\cdot 71^{6} + 54\cdot 71^{7} + 11\cdot 71^{8} +O(71^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 57 + 52\cdot 71 + 5\cdot 71^{2} + 18\cdot 71^{3} + 67\cdot 71^{4} + 15\cdot 71^{5} + 57\cdot 71^{6} + 28\cdot 71^{7} + 41\cdot 71^{8} +O(71^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 60 + 61\cdot 71 + 54\cdot 71^{2} + 32\cdot 71^{3} + 4\cdot 71^{4} + 41\cdot 71^{5} + 29\cdot 71^{6} + 15\cdot 71^{7} + 9\cdot 71^{8} +O(71^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 62 + 70\cdot 71 + 37\cdot 71^{2} + 63\cdot 71^{3} + 16\cdot 71^{4} + 29\cdot 71^{5} + 68\cdot 71^{6} + 58\cdot 71^{7} + 38\cdot 71^{8} +O(71^{9})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |