Basic invariants
Dimension: | $2$ |
Group: | $D_{8}$ |
Conductor: | \(855\)\(\medspace = 3^{2} \cdot 5 \cdot 19 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 8.0.1319500125.1 |
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 $\Q_{ 191 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 39 + 111\cdot 191 + 175\cdot 191^{2} + 190\cdot 191^{3} + 41\cdot 191^{4} +O(191^{5})\) |
$r_{ 2 }$ | $=$ | \( 54 + 71\cdot 191 + 122\cdot 191^{2} + 175\cdot 191^{3} + 108\cdot 191^{4} +O(191^{5})\) |
$r_{ 3 }$ | $=$ | \( 100 + 63\cdot 191 + 99\cdot 191^{2} + 63\cdot 191^{3} + 190\cdot 191^{4} +O(191^{5})\) |
$r_{ 4 }$ | $=$ | \( 114 + 149\cdot 191 + 64\cdot 191^{2} + 116\cdot 191^{4} +O(191^{5})\) |
$r_{ 5 }$ | $=$ | \( 139 + 2\cdot 191 + 128\cdot 191^{2} + 60\cdot 191^{3} + 54\cdot 191^{4} +O(191^{5})\) |
$r_{ 6 }$ | $=$ | \( 160 + 97\cdot 191 + 7\cdot 191^{2} + 97\cdot 191^{3} + 135\cdot 191^{4} +O(191^{5})\) |
$r_{ 7 }$ | $=$ | \( 163 + 9\cdot 191 + 105\cdot 191^{2} + 4\cdot 191^{3} + 101\cdot 191^{4} +O(191^{5})\) |
$r_{ 8 }$ | $=$ | \( 187 + 66\cdot 191 + 61\cdot 191^{2} + 171\cdot 191^{3} + 15\cdot 191^{4} +O(191^{5})\) |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$1$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-2$ | $-2$ |
$4$ | $2$ | $(1,5)(2,4)(6,8)$ | $0$ | $0$ |
$4$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ | $0$ |
$2$ | $4$ | $(1,4,2,5)(3,8,7,6)$ | $0$ | $0$ |
$2$ | $8$ | $(1,8,4,7,2,6,5,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
$2$ | $8$ | $(1,7,5,8,2,3,4,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |