Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| Artin number field: | Galois closure of 8.0.33973862400.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-5}, \sqrt{-6})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 83 + 56\cdot 83^{2} + 3\cdot 83^{3} + 61\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 29\cdot 83 + 64\cdot 83^{2} + 60\cdot 83^{3} + 24\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 + 27\cdot 83 + 33\cdot 83^{2} + 57\cdot 83^{3} + 81\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 38 + 2\cdot 83 + 21\cdot 83^{2} + 42\cdot 83^{3} + 76\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 + 80\cdot 83 + 61\cdot 83^{2} + 40\cdot 83^{3} + 6\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 62 + 55\cdot 83 + 49\cdot 83^{2} + 25\cdot 83^{3} + 83^{4} +O(83^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 70 + 53\cdot 83 + 18\cdot 83^{2} + 22\cdot 83^{3} + 58\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 78 + 81\cdot 83 + 26\cdot 83^{2} + 79\cdot 83^{3} + 21\cdot 83^{4} +O(83^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ | $0$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | $0$ |