Basic invariants
Dimension: | $2$ |
Group: | $Q_8:C_2$ |
Conductor: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
Artin stem field: | Galois closure of 8.0.191102976.3 |
Galois orbit size: | $2$ |
Smallest permutation container: | $Q_8:C_2$ |
Parity: | odd |
Determinant: | 1.8.2t1.b.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(i, \sqrt{6})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 6x^{6} + 18x^{4} - 36x^{2} + 36 \) . |
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 6 + 4\cdot 73 + 45\cdot 73^{2} + 4\cdot 73^{3} + 55\cdot 73^{4} +O(73^{5})\) |
$r_{ 2 }$ | $=$ | \( 13 + 21\cdot 73 + 47\cdot 73^{2} + 60\cdot 73^{3} + 39\cdot 73^{4} +O(73^{5})\) |
$r_{ 3 }$ | $=$ | \( 18 + 42\cdot 73 + 38\cdot 73^{2} + 48\cdot 73^{3} + 23\cdot 73^{4} +O(73^{5})\) |
$r_{ 4 }$ | $=$ | \( 34 + 17\cdot 73 + 42\cdot 73^{2} + 52\cdot 73^{3} + 62\cdot 73^{4} +O(73^{5})\) |
$r_{ 5 }$ | $=$ | \( 39 + 55\cdot 73 + 30\cdot 73^{2} + 20\cdot 73^{3} + 10\cdot 73^{4} +O(73^{5})\) |
$r_{ 6 }$ | $=$ | \( 55 + 30\cdot 73 + 34\cdot 73^{2} + 24\cdot 73^{3} + 49\cdot 73^{4} +O(73^{5})\) |
$r_{ 7 }$ | $=$ | \( 60 + 51\cdot 73 + 25\cdot 73^{2} + 12\cdot 73^{3} + 33\cdot 73^{4} +O(73^{5})\) |
$r_{ 8 }$ | $=$ | \( 67 + 68\cdot 73 + 27\cdot 73^{2} + 68\cdot 73^{3} + 17\cdot 73^{4} +O(73^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
$2$ | $2$ | $(2,7)(4,5)$ | $0$ |
$2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
$1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $-2 \zeta_{4}$ |
$2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
$2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
$2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.