Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(2475\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 11 \) |
| Artin number field: | Galois closure of 8.2.15160921875.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.2475.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 + 34\cdot 89 + 41\cdot 89^{2} + 48\cdot 89^{3} + 13\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 15\cdot 89 + 70\cdot 89^{2} + 32\cdot 89^{3} + 60\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 38 + 55\cdot 89 + 15\cdot 89^{2} + 35\cdot 89^{3} + 70\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 + 58\cdot 89 + 46\cdot 89^{2} + 69\cdot 89^{3} + 85\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 47 + 71\cdot 89 + 65\cdot 89^{2} + 17\cdot 89^{3} + 76\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 49 + 64\cdot 89 + 82\cdot 89^{2} + 65\cdot 89^{3} + 34\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 55 + 41\cdot 89^{2} + 75\cdot 89^{3} + 88\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 79 + 55\cdot 89 + 81\cdot 89^{2} + 10\cdot 89^{3} + 15\cdot 89^{4} +O(89^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,2)(4,8)(6,7)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,4,8)(3,6,5,7)$ | $0$ | $0$ |
| $4$ | $4$ | $(1,7,4,6)(2,5,8,3)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,6,2,5,4,7,8,3)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,7,2,3,4,6,8,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |