Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Artin number field: | Galois closure of 8.0.225815040000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-30}, \sqrt{-55})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 60 + 82\cdot 167 + 36\cdot 167^{2} + 82\cdot 167^{3} + 111\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 71 + 123\cdot 167 + 114\cdot 167^{2} + 90\cdot 167^{3} + 154\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 92 + 114\cdot 167 + 66\cdot 167^{2} + 155\cdot 167^{3} + 45\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 97 + 116\cdot 167 + 50\cdot 167^{2} + 92\cdot 167^{3} + 133\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 107 + 11\cdot 167 + 132\cdot 167^{2} + 68\cdot 167^{3} + 101\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 112 + 13\cdot 167 + 116\cdot 167^{2} + 5\cdot 167^{3} + 22\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 134 + 96\cdot 167 + 61\cdot 167^{2} + 100\cdot 167^{3} + 162\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 164 + 108\cdot 167 + 89\cdot 167^{2} + 72\cdot 167^{3} + 103\cdot 167^{4} +O(167^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,6)(4,8)(5,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $0$ | $0$ |
| $2$ | $2$ | $(3,6)(7,8)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,4,2,5)(3,8,6,7)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,2,4)(3,7,6,8)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,2,5)(3,7,6,8)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,6,2,3)(4,7,5,8)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,8,2,7)(3,4,6,5)$ | $0$ | $0$ |