Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(832\)\(\medspace = 2^{6} \cdot 13 \) |
| Artin stem field: | Galois closure of 8.0.575930368.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.13.4t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.35152.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{4} + 13 \)
|
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 19\cdot 53 + 2\cdot 53^{2} + 51\cdot 53^{3} + 45\cdot 53^{4} + 39\cdot 53^{5} + 42\cdot 53^{6} + 32\cdot 53^{7} + 43\cdot 53^{8} +O(53^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 16\cdot 53 + 17\cdot 53^{2} + 27\cdot 53^{3} + 24\cdot 53^{4} + 9\cdot 53^{5} + 4\cdot 53^{6} + 25\cdot 53^{7} + 12\cdot 53^{8} +O(53^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 7\cdot 53 + 45\cdot 53^{2} + 50\cdot 53^{3} + 51\cdot 53^{4} + 14\cdot 53^{5} + 24\cdot 53^{6} + 16\cdot 53^{7} + 41\cdot 53^{8} +O(53^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 4\cdot 53 + 28\cdot 53^{2} + 11\cdot 53^{3} + 23\cdot 53^{4} + 11\cdot 53^{5} + 4\cdot 53^{6} + 8\cdot 53^{7} + 48\cdot 53^{8} +O(53^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 + 48\cdot 53 + 24\cdot 53^{2} + 41\cdot 53^{3} + 29\cdot 53^{4} + 41\cdot 53^{5} + 48\cdot 53^{6} + 44\cdot 53^{7} + 4\cdot 53^{8} +O(53^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 + 45\cdot 53 + 7\cdot 53^{2} + 2\cdot 53^{3} + 53^{4} + 38\cdot 53^{5} + 28\cdot 53^{6} + 36\cdot 53^{7} + 11\cdot 53^{8} +O(53^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 31 + 36\cdot 53 + 35\cdot 53^{2} + 25\cdot 53^{3} + 28\cdot 53^{4} + 43\cdot 53^{5} + 48\cdot 53^{6} + 27\cdot 53^{7} + 40\cdot 53^{8} +O(53^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 38 + 33\cdot 53 + 50\cdot 53^{2} + 53^{3} + 7\cdot 53^{4} + 13\cdot 53^{5} + 10\cdot 53^{6} + 20\cdot 53^{7} + 9\cdot 53^{8} +O(53^{9})\)
|
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 value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ | |
| $4$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ | ✓ |
| $1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $-2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ | |
| $2$ | $4$ | $(2,6,7,3)$ | $\zeta_{4} + 1$ | |
| $2$ | $4$ | $(2,3,7,6)$ | $-\zeta_{4} + 1$ | |
| $2$ | $4$ | $(1,5,8,4)(2,7)(3,6)$ | $-\zeta_{4} - 1$ | |
| $2$ | $4$ | $(1,4,8,5)(2,7)(3,6)$ | $\zeta_{4} - 1$ | |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | |
| $4$ | $8$ | $(1,7,5,6,8,2,4,3)$ | $0$ | |
| $4$ | $8$ | $(1,6,4,7,8,3,5,2)$ | $0$ |