Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Artin number field: | Galois closure of 8.0.1194393600.9 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{15})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 5\cdot 61 + 58\cdot 61^{2} + 33\cdot 61^{3} + 16\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 18\cdot 61 + 35\cdot 61^{2} + 4\cdot 61^{3} + 9\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 24\cdot 61 + 20\cdot 61^{2} + 56\cdot 61^{3} + 3\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 60\cdot 61 + 9\cdot 61^{2} + 24\cdot 61^{3} + 51\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 32 + 51\cdot 61^{2} + 36\cdot 61^{3} + 9\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 37 + 36\cdot 61 + 40\cdot 61^{2} + 4\cdot 61^{3} + 57\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 45 + 42\cdot 61 + 25\cdot 61^{2} + 56\cdot 61^{3} + 51\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 48 + 55\cdot 61 + 2\cdot 61^{2} + 27\cdot 61^{3} + 44\cdot 61^{4} +O(61^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ | $0$ |