Basic invariants
Dimension: | $2$ |
Group: | $D_{8}$ |
Conductor: | \(548\)\(\medspace = 2^{2} \cdot 137 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.658266368.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{8}$ |
Parity: | odd |
Determinant: | 1.548.2t1.a.a |
Projective image: | $D_4$ |
Projective stem field: | Galois closure of 4.0.2192.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{7} + 12x^{6} - 18x^{5} + 17x^{4} - 6x^{3} + 2x^{2} + 12x + 4 \) . |
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 89\cdot 173 + 101\cdot 173^{2} + 74\cdot 173^{3} + 111\cdot 173^{4} + 115\cdot 173^{5} +O(173^{6})\) |
$r_{ 2 }$ | $=$ | \( 5 + 96\cdot 173 + 166\cdot 173^{2} + 35\cdot 173^{3} + 47\cdot 173^{4} + 42\cdot 173^{5} +O(173^{6})\) |
$r_{ 3 }$ | $=$ | \( 76 + 84\cdot 173^{2} + 166\cdot 173^{3} + 20\cdot 173^{4} + 25\cdot 173^{5} +O(173^{6})\) |
$r_{ 4 }$ | $=$ | \( 78 + 7\cdot 173 + 149\cdot 173^{2} + 127\cdot 173^{3} + 129\cdot 173^{4} + 124\cdot 173^{5} +O(173^{6})\) |
$r_{ 5 }$ | $=$ | \( 118 + 121\cdot 173 + 35\cdot 173^{2} + 61\cdot 173^{3} + 79\cdot 173^{4} + 153\cdot 173^{5} +O(173^{6})\) |
$r_{ 6 }$ | $=$ | \( 128 + 158\cdot 173 + 67\cdot 173^{2} + 25\cdot 173^{3} + 173^{4} + 161\cdot 173^{5} +O(173^{6})\) |
$r_{ 7 }$ | $=$ | \( 139 + 90\cdot 173 + 27\cdot 173^{2} + 118\cdot 173^{3} + 103\cdot 173^{4} + 117\cdot 173^{5} +O(173^{6})\) |
$r_{ 8 }$ | $=$ | \( 149 + 127\cdot 173 + 59\cdot 173^{2} + 82\cdot 173^{3} + 25\cdot 173^{4} + 125\cdot 173^{5} +O(173^{6})\) |
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 value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
$4$ | $2$ | $(1,8)(2,6)(3,7)(4,5)$ | $0$ |
$4$ | $2$ | $(1,4)(5,6)(7,8)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)(5,7,8,6)$ | $0$ |
$2$ | $8$ | $(1,7,3,8,4,6,2,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
$2$ | $8$ | $(1,8,2,7,4,5,3,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.