Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(2775\)\(\medspace = 3 \cdot 5^{2} \cdot 37 \) |
| Artin number field: | Galois closure of 8.0.1732640625.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{37})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 114\cdot 139 + 66\cdot 139^{2} + 9\cdot 139^{3} + 19\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 2\cdot 139 + 82\cdot 139^{2} + 134\cdot 139^{3} + 61\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 + 88\cdot 139 + 116\cdot 139^{2} + 29\cdot 139^{3} + 81\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 + 65\cdot 139 + 51\cdot 139^{2} + 40\cdot 139^{3} + 54\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 61 + 39\cdot 139 + 101\cdot 139^{2} + 60\cdot 139^{3} + 116\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 69 + 122\cdot 139 + 27\cdot 139^{2} + 73\cdot 139^{3} + 80\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 87 + 27\cdot 139 + 17\cdot 139^{2} + 84\cdot 139^{3} + 137\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 128 + 96\cdot 139 + 92\cdot 139^{2} + 123\cdot 139^{3} + 4\cdot 139^{4} +O(139^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $0$ | $0$ |
| $2$ | $2$ | $(2,6)(3,4)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,7,5,8)(2,3,6,4)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,5,7)(2,4,6,3)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,5,4)(2,7,6,8)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,7,5,8)(2,4,6,3)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,5,6)(3,8,4,7)$ | $0$ | $0$ |