Basic invariants
Dimension: | $2$ |
Group: | $D_{8}$ |
Conductor: | \(3136\)\(\medspace = 2^{6} \cdot 7^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.13492928512.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{8}$ |
Parity: | odd |
Determinant: | 1.4.2t1.a.a |
Projective image: | $D_4$ |
Projective stem field: | Galois closure of 4.0.1372.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 7x^{4} + 28 \) . |
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 11 + 5\cdot 53 + 21\cdot 53^{2} + 41\cdot 53^{3} + 10\cdot 53^{4} + 26\cdot 53^{5} +O(53^{6})\) |
$r_{ 2 }$ | $=$ | \( 12 + 46\cdot 53 + 53^{2} + 14\cdot 53^{3} + 2\cdot 53^{4} + 22\cdot 53^{5} +O(53^{6})\) |
$r_{ 3 }$ | $=$ | \( 13 + 14\cdot 53 + 34\cdot 53^{2} + 44\cdot 53^{3} + 12\cdot 53^{4} + 46\cdot 53^{5} +O(53^{6})\) |
$r_{ 4 }$ | $=$ | \( 19 + 32\cdot 53 + 8\cdot 53^{2} + 25\cdot 53^{3} + 34\cdot 53^{4} +O(53^{6})\) |
$r_{ 5 }$ | $=$ | \( 34 + 20\cdot 53 + 44\cdot 53^{2} + 27\cdot 53^{3} + 18\cdot 53^{4} + 52\cdot 53^{5} +O(53^{6})\) |
$r_{ 6 }$ | $=$ | \( 40 + 38\cdot 53 + 18\cdot 53^{2} + 8\cdot 53^{3} + 40\cdot 53^{4} + 6\cdot 53^{5} +O(53^{6})\) |
$r_{ 7 }$ | $=$ | \( 41 + 6\cdot 53 + 51\cdot 53^{2} + 38\cdot 53^{3} + 50\cdot 53^{4} + 30\cdot 53^{5} +O(53^{6})\) |
$r_{ 8 }$ | $=$ | \( 42 + 47\cdot 53 + 31\cdot 53^{2} + 11\cdot 53^{3} + 42\cdot 53^{4} + 26\cdot 53^{5} +O(53^{6})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$4$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
$4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ |
$2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
$2$ | $8$ | $(1,5,7,6,8,4,2,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
$2$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.