Basic invariants
Dimension: | $2$ |
Group: | $D_{8}$ |
Conductor: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 8.2.1438646272.4 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{8}$ |
Parity: | odd |
Projective image: | $D_4$ |
Projective field: | Galois closure of 4.0.1568.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 191 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 15 + 36\cdot 191 + 75\cdot 191^{2} + 188\cdot 191^{3} + 2\cdot 191^{4} + 130\cdot 191^{5} +O(191^{6})\) |
$r_{ 2 }$ | $=$ | \( 31 + 106\cdot 191 + 136\cdot 191^{2} + 165\cdot 191^{3} + 122\cdot 191^{4} + 97\cdot 191^{5} +O(191^{6})\) |
$r_{ 3 }$ | $=$ | \( 33 + 146\cdot 191 + 90\cdot 191^{2} + 42\cdot 191^{3} + 73\cdot 191^{4} + 59\cdot 191^{5} +O(191^{6})\) |
$r_{ 4 }$ | $=$ | \( 52 + 187\cdot 191 + 59\cdot 191^{2} + 53\cdot 191^{3} + 81\cdot 191^{4} + 105\cdot 191^{5} +O(191^{6})\) |
$r_{ 5 }$ | $=$ | \( 69 + 6\cdot 191 + 81\cdot 191^{2} + 116\cdot 191^{3} + 52\cdot 191^{4} + 41\cdot 191^{5} +O(191^{6})\) |
$r_{ 6 }$ | $=$ | \( 95 + 52\cdot 191 + 110\cdot 191^{2} + 165\cdot 191^{3} + 174\cdot 191^{4} + 48\cdot 191^{5} +O(191^{6})\) |
$r_{ 7 }$ | $=$ | \( 120 + 45\cdot 191 + 167\cdot 191^{2} + 76\cdot 191^{3} + 178\cdot 191^{4} + 19\cdot 191^{5} +O(191^{6})\) |
$r_{ 8 }$ | $=$ | \( 162 + 183\cdot 191 + 42\cdot 191^{2} + 146\cdot 191^{3} + 77\cdot 191^{4} + 70\cdot 191^{5} +O(191^{6})\) |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-2$ | $-2$ |
$4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ | $0$ |
$4$ | $2$ | $(1,6)(2,4)(7,8)$ | $0$ | $0$ |
$2$ | $4$ | $(1,4,2,6)(3,8,5,7)$ | $0$ | $0$ |
$2$ | $8$ | $(1,5,6,8,2,3,4,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
$2$ | $8$ | $(1,8,4,5,2,7,6,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |