Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3648\)\(\medspace = 2^{6} \cdot 3 \cdot 19 \) |
| Artin number field: | Galois closure of 8.0.1916338176.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-38})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 34\cdot 109 + 6\cdot 109^{2} + 3\cdot 109^{3} + 20\cdot 109^{4} + 39\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 54\cdot 109 + 17\cdot 109^{2} + 99\cdot 109^{3} + 103\cdot 109^{4} + 45\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 + 100\cdot 109 + 30\cdot 109^{2} + 31\cdot 109^{3} + 91\cdot 109^{4} + 61\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 45 + 5\cdot 109 + 83\cdot 109^{2} + 83\cdot 109^{3} + 15\cdot 109^{4} + 38\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 64 + 103\cdot 109 + 25\cdot 109^{2} + 25\cdot 109^{3} + 93\cdot 109^{4} + 70\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 79 + 8\cdot 109 + 78\cdot 109^{2} + 77\cdot 109^{3} + 17\cdot 109^{4} + 47\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 86 + 54\cdot 109 + 91\cdot 109^{2} + 9\cdot 109^{3} + 5\cdot 109^{4} + 63\cdot 109^{5} +O(109^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 103 + 74\cdot 109 + 102\cdot 109^{2} + 105\cdot 109^{3} + 88\cdot 109^{4} + 69\cdot 109^{5} +O(109^{6})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ | $0$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ | $0$ |