Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(395\)\(\medspace = 5 \cdot 79 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.308149375.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.395.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.1975.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 4x^{6} - 7x^{5} + 8x^{4} - 11x^{3} + 11x^{2} - 2x - 4 \)
|
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 + 4\cdot 101 + 5\cdot 101^{2} + 95\cdot 101^{3} + 33\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 71\cdot 101 + 97\cdot 101^{2} + 88\cdot 101^{3} + 51\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 + 62\cdot 101 + 31\cdot 101^{2} + 79\cdot 101^{3} + 63\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 38 + 77\cdot 101 + 26\cdot 101^{2} + 72\cdot 101^{3} + 3\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 46 + 5\cdot 101 + 66\cdot 101^{2} + 32\cdot 101^{3} + 93\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 53 + 87\cdot 101 + 93\cdot 101^{2} + 47\cdot 101^{3} + 19\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 89 + 101 + 44\cdot 101^{2} + 98\cdot 101^{3} + 80\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 92 + 93\cdot 101 + 38\cdot 101^{2} + 91\cdot 101^{3} + 56\cdot 101^{4} +O(101^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,8)(6,7)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,8,2,4)(3,6,5,7)$ | $0$ |
| $2$ | $8$ | $(1,5,4,6,2,3,8,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,6,8,5,2,7,4,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.