Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(855\)\(\medspace = 3^{2} \cdot 5 \cdot 19 \) |
| Artin number field: | Galois closure of 8.0.164480625.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-15}, \sqrt{-19})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 48\cdot 61 + 55\cdot 61^{2} + 48\cdot 61^{3} + 18\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 40\cdot 61 + 55\cdot 61^{2} + 46\cdot 61^{3} + 40\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 + 16\cdot 61 + 9\cdot 61^{2} + 53\cdot 61^{3} + 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 + 39\cdot 61 + 19\cdot 61^{2} + 21\cdot 61^{3} + 19\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 42 + 24\cdot 61 + 61^{2} + 56\cdot 61^{3} + 4\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 45 + 21\cdot 61 + 11\cdot 61^{2} + 32\cdot 61^{3} + 16\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 57 + 43\cdot 61 + 26\cdot 61^{2} + 13\cdot 61^{3} + 58\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 60 + 9\cdot 61 + 3\cdot 61^{2} + 33\cdot 61^{3} + 22\cdot 61^{4} +O(61^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $0$ | $0$ |
| $2$ | $2$ | $(4,6)(5,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,7)(2,5)(3,4)(6,8)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,8,2,3)(4,7,6,5)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,2,8)(4,5,6,7)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,2,5)(3,4,8,6)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,6,2,4)(3,7,8,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,8,2,3)(4,5,6,7)$ | $0$ | $0$ |