Basic invariants
Dimension: | $2$ |
Group: | $Q_8:C_2$ |
Conductor: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
Artin stem field: | Galois closure of 8.0.33973862400.4 |
Galois orbit size: | $2$ |
Smallest permutation container: | $Q_8:C_2$ |
Parity: | odd |
Determinant: | 1.15.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-5}, \sqrt{-6})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{6} + 12x^{4} + 20x^{2} + 25 \) . |
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 5 + 83 + 56\cdot 83^{2} + 3\cdot 83^{3} + 61\cdot 83^{4} +O(83^{5})\) |
$r_{ 2 }$ | $=$ | \( 13 + 29\cdot 83 + 64\cdot 83^{2} + 60\cdot 83^{3} + 24\cdot 83^{4} +O(83^{5})\) |
$r_{ 3 }$ | $=$ | \( 21 + 27\cdot 83 + 33\cdot 83^{2} + 57\cdot 83^{3} + 81\cdot 83^{4} +O(83^{5})\) |
$r_{ 4 }$ | $=$ | \( 38 + 2\cdot 83 + 21\cdot 83^{2} + 42\cdot 83^{3} + 76\cdot 83^{4} +O(83^{5})\) |
$r_{ 5 }$ | $=$ | \( 45 + 80\cdot 83 + 61\cdot 83^{2} + 40\cdot 83^{3} + 6\cdot 83^{4} +O(83^{5})\) |
$r_{ 6 }$ | $=$ | \( 62 + 55\cdot 83 + 49\cdot 83^{2} + 25\cdot 83^{3} + 83^{4} +O(83^{5})\) |
$r_{ 7 }$ | $=$ | \( 70 + 53\cdot 83 + 18\cdot 83^{2} + 22\cdot 83^{3} + 58\cdot 83^{4} +O(83^{5})\) |
$r_{ 8 }$ | $=$ | \( 78 + 81\cdot 83 + 26\cdot 83^{2} + 79\cdot 83^{3} + 21\cdot 83^{4} +O(83^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
$2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
$2$ | $2$ | $(2,7)(4,5)$ | $0$ |
$1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $-2 \zeta_{4}$ |
$2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
$2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
$2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.