Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(95\)\(\medspace = 5 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.2.4286875.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.475.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 129\cdot 131 + 3\cdot 131^{2} + 118\cdot 131^{3} + 70\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 112\cdot 131 + 72\cdot 131^{2} + 124\cdot 131^{3} + 36\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 + 14\cdot 131 + 37\cdot 131^{2} + 104\cdot 131^{3} + 5\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 68 + 75\cdot 131 + 38\cdot 131^{2} + 30\cdot 131^{3} + 25\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 72 + 104\cdot 131 + 34\cdot 131^{2} + 104\cdot 131^{3} + 68\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 86 + 16\cdot 131 + 87\cdot 131^{2} + 94\cdot 131^{3} + 60\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 114 + 85\cdot 131 + 91\cdot 131^{2} + 84\cdot 131^{3} + 14\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 124 + 116\cdot 131 + 26\cdot 131^{2} + 125\cdot 131^{3} + 109\cdot 131^{4} +O(131^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(2,8)(3,4)(5,6)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,6,7,5)(2,3,4,8)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,2,6,3,7,4,5,8)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,5,2,7,8,6,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |