Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(2120\)\(\medspace = 2^{3} \cdot 5 \cdot 53 \) |
| Artin number field: | Galois closure of 8.0.7191040000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-10}, \sqrt{-106})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 211 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 27 + 70\cdot 211 + 17\cdot 211^{2} + 15\cdot 211^{3} + 86\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 + 159\cdot 211 + 66\cdot 211^{2} + 50\cdot 211^{3} + 204\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 34 + 6\cdot 211 + 163\cdot 211^{2} + 182\cdot 211^{3} + 130\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 + 64\cdot 211 + 66\cdot 211^{2} + 146\cdot 211^{3} + 69\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 95 + 191\cdot 211 + 102\cdot 211^{2} + 129\cdot 211^{3} + 116\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 105 + 30\cdot 211 + 57\cdot 211^{2} + 142\cdot 211^{3} + 6\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 127 + 119\cdot 211 + 89\cdot 211^{2} + 153\cdot 211^{3} + 92\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 174 + 202\cdot 211 + 69\cdot 211^{2} + 24\cdot 211^{3} + 137\cdot 211^{4} +O(211^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(2,7)(3,5)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,4,8,6)(2,5,7,3)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,4)(2,3,7,5)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,8,3)(2,4,7,6)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,4,8,6)(2,3,7,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,5,6)$ | $0$ | $0$ |