Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1220\)\(\medspace = 2^{2} \cdot 5 \cdot 61 \) |
| Artin number field: | Galois closure of 8.0.595360000.5 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-5}, \sqrt{-61})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 21\cdot 41 + 5\cdot 41^{2} + 41^{3} + 21\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 20\cdot 41 + 37\cdot 41^{2} + 34\cdot 41^{3} + 26\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 28\cdot 41 + 8\cdot 41^{2} + 13\cdot 41^{3} + 18\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 6\cdot 41 + 25\cdot 41^{2} + 13\cdot 41^{3} + 14\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 + 12\cdot 41 + 30\cdot 41^{2} + 32\cdot 41^{3} + 15\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 31 + 29\cdot 41 + 31\cdot 41^{2} + 17\cdot 41^{3} + 3\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 32 + 27\cdot 41 + 7\cdot 41^{2} + 12\cdot 41^{3} + 25\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 34 + 17\cdot 41 + 17\cdot 41^{2} + 38\cdot 41^{3} + 38\cdot 41^{4} +O(41^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ | $0$ |
| $2$ | $2$ | $(2,3)(6,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,8,5,4)(2,6,3,7)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,5,8)(2,7,3,6)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,5,7)(2,8,3,4)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,3,5,2)(4,6,8,7)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,8,5,4)(2,7,3,6)$ | $0$ | $0$ |