Basic invariants
Dimension: | $2$ |
Group: | $Q_8:C_2$ |
Conductor: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
Artin stem field: | Galois closure of 8.4.2621440000.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $Q_8:C_2$ |
Parity: | odd |
Determinant: | 1.20.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(i, \sqrt{10})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 6x^{4} + 4 \) . |
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 27\cdot 41 + 15\cdot 41^{2} + 27\cdot 41^{3} + 8\cdot 41^{4} + 10\cdot 41^{5} +O(41^{6})\) |
$r_{ 2 }$ | $=$ | \( 12 + 41 + 30\cdot 41^{2} + 17\cdot 41^{3} + 7\cdot 41^{4} + 8\cdot 41^{5} +O(41^{6})\) |
$r_{ 3 }$ | $=$ | \( 15 + 3\cdot 41 + 7\cdot 41^{2} + 35\cdot 41^{4} + 5\cdot 41^{5} +O(41^{6})\) |
$r_{ 4 }$ | $=$ | \( 18 + 15\cdot 41 + 41^{2} + 39\cdot 41^{3} + 23\cdot 41^{4} + 2\cdot 41^{5} +O(41^{6})\) |
$r_{ 5 }$ | $=$ | \( 23 + 25\cdot 41 + 39\cdot 41^{2} + 41^{3} + 17\cdot 41^{4} + 38\cdot 41^{5} +O(41^{6})\) |
$r_{ 6 }$ | $=$ | \( 26 + 37\cdot 41 + 33\cdot 41^{2} + 40\cdot 41^{3} + 5\cdot 41^{4} + 35\cdot 41^{5} +O(41^{6})\) |
$r_{ 7 }$ | $=$ | \( 29 + 39\cdot 41 + 10\cdot 41^{2} + 23\cdot 41^{3} + 33\cdot 41^{4} + 32\cdot 41^{5} +O(41^{6})\) |
$r_{ 8 }$ | $=$ | \( 39 + 13\cdot 41 + 25\cdot 41^{2} + 13\cdot 41^{3} + 32\cdot 41^{4} + 30\cdot 41^{5} +O(41^{6})\) |
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 | Complex conjugation |
$1$ | $1$ | $()$ | $2$ | |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
$2$ | $2$ | $(3,6)(4,5)$ | $0$ | ✓ |
$2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ | |
$2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ | |
$1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $-2 \zeta_{4}$ | |
$1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $2 \zeta_{4}$ | |
$2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ | |
$2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ | |
$2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |