Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(2280\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 19 \) |
| Artin number field: | Galois closure of 8.0.46915560000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{30}, \sqrt{-95})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 97\cdot 101 + 95\cdot 101^{2} + 88\cdot 101^{3} + 23\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 10\cdot 101 + 6\cdot 101^{2} + 59\cdot 101^{3} + 61\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 + 60\cdot 101 + 80\cdot 101^{2} + 25\cdot 101^{3} + 10\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 35 + 20\cdot 101 + 32\cdot 101^{2} + 5\cdot 101^{3} + 70\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 66 + 80\cdot 101 + 68\cdot 101^{2} + 95\cdot 101^{3} + 30\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 88 + 40\cdot 101 + 20\cdot 101^{2} + 75\cdot 101^{3} + 90\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 97 + 90\cdot 101 + 94\cdot 101^{2} + 41\cdot 101^{3} + 39\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 98 + 3\cdot 101 + 5\cdot 101^{2} + 12\cdot 101^{3} + 77\cdot 101^{4} +O(101^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | $0$ |