Basic invariants
Dimension: | $2$ |
Group: | $Q_8$ |
Conductor: | \(665856\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 17^{2} \) |
Frobenius-Schur indicator: | $-1$ |
Root number: | $-1$ |
Artin field: | Galois closure of 8.0.1021511146143744.1 |
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 \(\Q(\sqrt{2}, \sqrt{3})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} + 204x^{6} + 10404x^{4} + 176868x^{2} + 751689 \) . |
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 10.
Roots:
$r_{ 1 }$ | $=$ | \( 5 + 8\cdot 23 + 19\cdot 23^{2} + 17\cdot 23^{3} + 16\cdot 23^{4} + 19\cdot 23^{5} + 3\cdot 23^{6} + 6\cdot 23^{8} + 14\cdot 23^{9} +O(23^{10})\) |
$r_{ 2 }$ | $=$ | \( 6 + 8\cdot 23 + 8\cdot 23^{2} + 21\cdot 23^{3} + 19\cdot 23^{5} + 7\cdot 23^{6} + 5\cdot 23^{7} + 17\cdot 23^{8} + 8\cdot 23^{9} +O(23^{10})\) |
$r_{ 3 }$ | $=$ | \( 7 + 14\cdot 23 + 12\cdot 23^{2} + 16\cdot 23^{3} + 18\cdot 23^{4} + 15\cdot 23^{5} + 6\cdot 23^{6} + 8\cdot 23^{7} + 4\cdot 23^{8} + 10\cdot 23^{9} +O(23^{10})\) |
$r_{ 4 }$ | $=$ | \( 10 + 15\cdot 23 + 7\cdot 23^{2} + 14\cdot 23^{3} + 5\cdot 23^{4} + 22\cdot 23^{5} + 12\cdot 23^{6} + 19\cdot 23^{7} + 18\cdot 23^{8} +O(23^{10})\) |
$r_{ 5 }$ | $=$ | \( 13 + 7\cdot 23 + 15\cdot 23^{2} + 8\cdot 23^{3} + 17\cdot 23^{4} + 10\cdot 23^{6} + 3\cdot 23^{7} + 4\cdot 23^{8} + 22\cdot 23^{9} +O(23^{10})\) |
$r_{ 6 }$ | $=$ | \( 16 + 8\cdot 23 + 10\cdot 23^{2} + 6\cdot 23^{3} + 4\cdot 23^{4} + 7\cdot 23^{5} + 16\cdot 23^{6} + 14\cdot 23^{7} + 18\cdot 23^{8} + 12\cdot 23^{9} +O(23^{10})\) |
$r_{ 7 }$ | $=$ | \( 17 + 14\cdot 23 + 14\cdot 23^{2} + 23^{3} + 22\cdot 23^{4} + 3\cdot 23^{5} + 15\cdot 23^{6} + 17\cdot 23^{7} + 5\cdot 23^{8} + 14\cdot 23^{9} +O(23^{10})\) |
$r_{ 8 }$ | $=$ | \( 18 + 14\cdot 23 + 3\cdot 23^{2} + 5\cdot 23^{3} + 6\cdot 23^{4} + 3\cdot 23^{5} + 19\cdot 23^{6} + 22\cdot 23^{7} + 16\cdot 23^{8} + 8\cdot 23^{9} +O(23^{10})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
$2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
$2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.