Basic invariants
Dimension: | $2$ |
Group: | $Q_8$ |
Conductor: | \(278784\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 11^{2} \) |
Frobenius-Schur indicator: | $-1$ |
Root number: | $-1$ |
Artin field: | Galois closure of 8.0.21667237072994304.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $Q_8$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{6}, \sqrt{11})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} + 132x^{6} + 5940x^{4} + 100188x^{2} + 393129 \) . |
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 10.
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 40\cdot 43 + 31\cdot 43^{2} + 41\cdot 43^{3} + 19\cdot 43^{4} + 29\cdot 43^{5} + 38\cdot 43^{8} + 7\cdot 43^{9} +O(43^{10})\) |
$r_{ 2 }$ | $=$ | \( 5 + 19\cdot 43 + 7\cdot 43^{2} + 5\cdot 43^{3} + 8\cdot 43^{4} + 19\cdot 43^{5} + 3\cdot 43^{6} + 17\cdot 43^{7} + 30\cdot 43^{9} +O(43^{10})\) |
$r_{ 3 }$ | $=$ | \( 16 + 12\cdot 43 + 12\cdot 43^{2} + 43^{3} + 25\cdot 43^{4} + 36\cdot 43^{5} + 18\cdot 43^{6} + 43^{7} + 11\cdot 43^{8} + 3\cdot 43^{9} +O(43^{10})\) |
$r_{ 4 }$ | $=$ | \( 20 + 36\cdot 43 + 39\cdot 43^{2} + 23\cdot 43^{3} + 43^{4} + 12\cdot 43^{5} + 10\cdot 43^{6} + 12\cdot 43^{7} + 35\cdot 43^{8} + 35\cdot 43^{9} +O(43^{10})\) |
$r_{ 5 }$ | $=$ | \( 23 + 6\cdot 43 + 3\cdot 43^{2} + 19\cdot 43^{3} + 41\cdot 43^{4} + 30\cdot 43^{5} + 32\cdot 43^{6} + 30\cdot 43^{7} + 7\cdot 43^{8} + 7\cdot 43^{9} +O(43^{10})\) |
$r_{ 6 }$ | $=$ | \( 27 + 30\cdot 43 + 30\cdot 43^{2} + 41\cdot 43^{3} + 17\cdot 43^{4} + 6\cdot 43^{5} + 24\cdot 43^{6} + 41\cdot 43^{7} + 31\cdot 43^{8} + 39\cdot 43^{9} +O(43^{10})\) |
$r_{ 7 }$ | $=$ | \( 38 + 23\cdot 43 + 35\cdot 43^{2} + 37\cdot 43^{3} + 34\cdot 43^{4} + 23\cdot 43^{5} + 39\cdot 43^{6} + 25\cdot 43^{7} + 42\cdot 43^{8} + 12\cdot 43^{9} +O(43^{10})\) |
$r_{ 8 }$ | $=$ | \( 41 + 2\cdot 43 + 11\cdot 43^{2} + 43^{3} + 23\cdot 43^{4} + 13\cdot 43^{5} + 42\cdot 43^{6} + 42\cdot 43^{7} + 4\cdot 43^{8} + 35\cdot 43^{9} +O(43^{10})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
$2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
$2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.