Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1560\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 13 \) |
| Artin stem field: | Galois closure of 8.0.3701505600.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.1560.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-30}, \sqrt{-39})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{6} - 21x^{5} + 23x^{4} + 126x^{3} + 198x^{2} + 156x + 52 \)
|
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 95\cdot 157 + 120\cdot 157^{2} + 75\cdot 157^{3} + 70\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 + 87\cdot 157 + 98\cdot 157^{2} + 13\cdot 157^{3} + 149\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 55 + 63\cdot 157 + 147\cdot 157^{2} + 84\cdot 157^{3} + 100\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 65 + 68\cdot 157 + 104\cdot 157^{2} + 139\cdot 157^{3} + 150\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 96 + 42\cdot 157 + 143\cdot 157^{2} + 4\cdot 157^{3} + 41\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 106 + 102\cdot 157 + 82\cdot 157^{2} + 126\cdot 157^{3} + 64\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 116 + 38\cdot 157 + 138\cdot 157^{2} + 16\cdot 157^{3} + 148\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 153 + 129\cdot 157 + 106\cdot 157^{2} + 8\cdot 157^{3} + 60\cdot 157^{4} +O(157^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,5)(3,7)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $0$ |
| $2$ | $2$ | $(5,7)(6,8)$ | $0$ |
| $1$ | $4$ | $(1,3,4,2)(5,8,7,6)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)(5,6,7,8)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,4,8)(2,7,3,5)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)(5,6,7,8)$ | $0$ |
| $2$ | $4$ | $(1,5,4,7)(2,6,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.