Basic invariants
Dimension: | $2$ |
Group: | $D_{8}$ |
Conductor: | \(2004\)\(\medspace = 2^{2} \cdot 3 \cdot 167 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.2.96577152768.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{8}$ |
Parity: | odd |
Determinant: | 1.2004.2t1.a.a |
Projective image: | $D_4$ |
Projective stem field: | Galois closure of 4.2.24048.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 11x^{6} + 64x^{4} - 60x^{3} - 15x^{2} - 36x + 9 \) . |
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 6 + 112\cdot 179 + 43\cdot 179^{3} + 106\cdot 179^{4} + 17\cdot 179^{5} +O(179^{6})\) |
$r_{ 2 }$ | $=$ | \( 23 + 40\cdot 179 + 85\cdot 179^{2} + 30\cdot 179^{3} + 29\cdot 179^{4} + 110\cdot 179^{5} +O(179^{6})\) |
$r_{ 3 }$ | $=$ | \( 35 + 131\cdot 179 + 21\cdot 179^{2} + 119\cdot 179^{3} + 66\cdot 179^{4} + 122\cdot 179^{5} +O(179^{6})\) |
$r_{ 4 }$ | $=$ | \( 86 + 47\cdot 179 + 67\cdot 179^{2} + 26\cdot 179^{3} + 131\cdot 179^{4} + 34\cdot 179^{5} +O(179^{6})\) |
$r_{ 5 }$ | $=$ | \( 96 + 140\cdot 179 + 157\cdot 179^{2} + 153\cdot 179^{3} + 93\cdot 179^{4} + 140\cdot 179^{5} +O(179^{6})\) |
$r_{ 6 }$ | $=$ | \( 135 + 8\cdot 179 + 158\cdot 179^{2} + 164\cdot 179^{3} + 108\cdot 179^{4} + 111\cdot 179^{5} +O(179^{6})\) |
$r_{ 7 }$ | $=$ | \( 166 + 154\cdot 179 + 163\cdot 179^{2} + 125\cdot 179^{3} + 70\cdot 179^{4} + 116\cdot 179^{5} +O(179^{6})\) |
$r_{ 8 }$ | $=$ | \( 169 + 80\cdot 179 + 61\cdot 179^{2} + 52\cdot 179^{3} + 109\cdot 179^{4} + 62\cdot 179^{5} +O(179^{6})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,2)(3,5)(4,8)(6,7)$ | $-2$ |
$4$ | $2$ | $(1,2)(4,7)(6,8)$ | $0$ |
$4$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $0$ |
$2$ | $4$ | $(1,3,2,5)(4,6,8,7)$ | $0$ |
$2$ | $8$ | $(1,6,3,8,2,7,5,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
$2$ | $8$ | $(1,8,5,6,2,4,3,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.