Basic invariants
Dimension: | $2$ |
Group: | $Q_8:C_2$ |
Conductor: | \(3520\)\(\medspace = 2^{6} \cdot 5 \cdot 11 \) |
Artin stem field: | Galois closure of 8.0.23987814400.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $Q_8:C_2$ |
Parity: | odd |
Determinant: | 1.440.2t1.b.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-10}, \sqrt{-11})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 20x^{6} + 101x^{4} - 10x^{2} + 25 \) . |
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 28 + 153\cdot 157 + 116\cdot 157^{2} + 55\cdot 157^{3} + 88\cdot 157^{4} + 42\cdot 157^{5} +O(157^{6})\) |
$r_{ 2 }$ | $=$ | \( 47 + 74\cdot 157 + 147\cdot 157^{2} + 63\cdot 157^{3} + 15\cdot 157^{4} + 34\cdot 157^{5} +O(157^{6})\) |
$r_{ 3 }$ | $=$ | \( 53 + 59\cdot 157 + 146\cdot 157^{2} + 139\cdot 157^{3} + 149\cdot 157^{4} + 157^{5} +O(157^{6})\) |
$r_{ 4 }$ | $=$ | \( 59 + 101\cdot 157 + 26\cdot 157^{2} + 98\cdot 157^{3} + 17\cdot 157^{4} + 87\cdot 157^{5} +O(157^{6})\) |
$r_{ 5 }$ | $=$ | \( 98 + 55\cdot 157 + 130\cdot 157^{2} + 58\cdot 157^{3} + 139\cdot 157^{4} + 69\cdot 157^{5} +O(157^{6})\) |
$r_{ 6 }$ | $=$ | \( 104 + 97\cdot 157 + 10\cdot 157^{2} + 17\cdot 157^{3} + 7\cdot 157^{4} + 155\cdot 157^{5} +O(157^{6})\) |
$r_{ 7 }$ | $=$ | \( 110 + 82\cdot 157 + 9\cdot 157^{2} + 93\cdot 157^{3} + 141\cdot 157^{4} + 122\cdot 157^{5} +O(157^{6})\) |
$r_{ 8 }$ | $=$ | \( 129 + 3\cdot 157 + 40\cdot 157^{2} + 101\cdot 157^{3} + 68\cdot 157^{4} + 114\cdot 157^{5} +O(157^{6})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
$2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
$2$ | $2$ | $(1,8)(2,7)$ | $0$ |
$1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $-2 \zeta_{4}$ |
$1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $2 \zeta_{4}$ |
$2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
$2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
$2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.