Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Artin number field: | Galois closure of 8.0.4665600.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{15})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 84\cdot 109 + 49\cdot 109^{2} + 42\cdot 109^{3} + 38\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 63\cdot 109 + 17\cdot 109^{2} + 87\cdot 109^{3} + 75\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 81\cdot 109 + 92\cdot 109^{2} + 9\cdot 109^{3} + 93\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 67 + 2\cdot 109 + 59\cdot 109^{2} + 90\cdot 109^{3} + 10\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 73 + 70\cdot 109 + 42\cdot 109^{2} + 104\cdot 109^{3} + 2\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 75 + 102\cdot 109 + 35\cdot 109^{2} + 67\cdot 109^{3} + 85\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 90 + 37\cdot 109 + 107\cdot 109^{2} + 36\cdot 109^{3} + 35\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 99 + 102\cdot 109 + 30\cdot 109^{2} + 106\cdot 109^{3} + 93\cdot 109^{4} +O(109^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $0$ | $0$ |
| $2$ | $2$ | $(2,5)(6,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,3,4,8)(2,6,5,7)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,4,3)(2,7,5,6)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,4,7)(2,3,5,8)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,3,4,8)(2,7,5,6)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,5,4,2)(3,7,8,6)$ | $0$ | $0$ |