Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(825\)\(\medspace = 3 \cdot 5^{2} \cdot 11 \) |
| Artin number field: | Galois closure of 8.0.16471125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.12375.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 24\cdot 59 + 48\cdot 59^{2} + 5\cdot 59^{3} + 40\cdot 59^{4} + 32\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 35\cdot 59 + 21\cdot 59^{2} + 42\cdot 59^{3} + 18\cdot 59^{4} + 27\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 + 44\cdot 59 + 57\cdot 59^{2} + 7\cdot 59^{3} + 11\cdot 59^{4} + 37\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 + 3\cdot 59 + 29\cdot 59^{2} + 45\cdot 59^{3} + 19\cdot 59^{4} + 11\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 + 23\cdot 59 + 30\cdot 59^{2} + 28\cdot 59^{3} + 39\cdot 59^{4} + 57\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 42 + 36\cdot 59 + 24\cdot 59^{2} + 48\cdot 59^{3} + 11\cdot 59^{4} + 13\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 48 + 38\cdot 59 + 10\cdot 59^{2} + 54\cdot 59^{3} + 2\cdot 59^{4} + 39\cdot 59^{5} +O(59^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 52 + 29\cdot 59 + 13\cdot 59^{2} + 3\cdot 59^{3} + 33\cdot 59^{4} + 17\cdot 59^{5} +O(59^{6})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,4)(6,8)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(2,7)(3,4)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,6,5,8)(2,4,7,3)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,5,6)(2,3,7,4)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(2,3,7,4)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(2,4,7,3)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,5)(2,4,7,3)(6,8)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,5)(2,3,7,4)(6,8)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,5,8)(2,3,7,4)$ | $0$ | $0$ |
| $4$ | $4$ | $(1,4,5,3)(2,6,7,8)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,2,6,4,5,7,8,3)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,4,8,2,5,3,6,7)$ | $0$ | $0$ |