Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3380\)\(\medspace = 2^{2} \cdot 5 \cdot 13^{2} \) |
| Artin number field: | Galois closure of 8.0.30891577600.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{65})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 25\cdot 181 + 19\cdot 181^{2} + 160\cdot 181^{3} + 77\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 41\cdot 181 + 77\cdot 181^{2} + 113\cdot 181^{3} + 81\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 45 + 89\cdot 181 + 62\cdot 181^{2} + 97\cdot 181^{3} + 74\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 51 + 40\cdot 181 + 156\cdot 181^{2} + 41\cdot 181^{3} + 54\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 59 + 38\cdot 181 + 179\cdot 181^{2} + 15\cdot 181^{3} + 88\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 91 + 47\cdot 181 + 39\cdot 181^{2} + 136\cdot 181^{3} + 91\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 97 + 177\cdot 181 + 111\cdot 181^{2} + 118\cdot 181^{3} + 114\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 173 + 83\cdot 181 + 78\cdot 181^{2} + 40\cdot 181^{3} + 141\cdot 181^{4} +O(181^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,6)(3,4)(5,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $0$ | $0$ |
| $2$ | $2$ | $(2,8)(4,7)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,5,6,3)(2,4,8,7)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,6,5)(2,7,8,4)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,6,3)(2,7,8,4)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,6,8)(3,7,5,4)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,7,6,4)(2,3,8,5)$ | $0$ | $0$ |