Basic invariants
Dimension: | $2$ |
Group: | $Q_8:C_2$ |
Conductor: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
Artin number field: | Galois closure of 8.4.33973862400.4 |
Galois orbit size: | $2$ |
Smallest permutation container: | $Q_8:C_2$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-5}, \sqrt{6})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 7.
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 16\cdot 23 + 13\cdot 23^{2} + 9\cdot 23^{3} + 5\cdot 23^{4} + 7\cdot 23^{5} + 18\cdot 23^{6} +O(23^{7})\) |
$r_{ 2 }$ | $=$ | \( 7 + 3\cdot 23 + 15\cdot 23^{3} + 2\cdot 23^{4} + 18\cdot 23^{5} + 10\cdot 23^{6} +O(23^{7})\) |
$r_{ 3 }$ | $=$ | \( 9 + 22\cdot 23 + 23^{2} + 15\cdot 23^{3} + 3\cdot 23^{4} + 5\cdot 23^{5} + 3\cdot 23^{6} +O(23^{7})\) |
$r_{ 4 }$ | $=$ | \( 10 + 2\cdot 23 + 14\cdot 23^{2} + 4\cdot 23^{4} + 12\cdot 23^{5} + 7\cdot 23^{6} +O(23^{7})\) |
$r_{ 5 }$ | $=$ | \( 13 + 20\cdot 23 + 8\cdot 23^{2} + 22\cdot 23^{3} + 18\cdot 23^{4} + 10\cdot 23^{5} + 15\cdot 23^{6} +O(23^{7})\) |
$r_{ 6 }$ | $=$ | \( 14 + 21\cdot 23^{2} + 7\cdot 23^{3} + 19\cdot 23^{4} + 17\cdot 23^{5} + 19\cdot 23^{6} +O(23^{7})\) |
$r_{ 7 }$ | $=$ | \( 16 + 19\cdot 23 + 22\cdot 23^{2} + 7\cdot 23^{3} + 20\cdot 23^{4} + 4\cdot 23^{5} + 12\cdot 23^{6} +O(23^{7})\) |
$r_{ 8 }$ | $=$ | \( 21 + 6\cdot 23 + 9\cdot 23^{2} + 13\cdot 23^{3} + 17\cdot 23^{4} + 15\cdot 23^{5} + 4\cdot 23^{6} +O(23^{7})\) |
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 values | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
$2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ | $0$ |
$2$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ | $0$ |
$2$ | $2$ | $(2,7)(3,6)$ | $0$ | $0$ |
$1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
$2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ | $0$ |
$2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ | $0$ |
$2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | $0$ |