Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(1664\)\(\medspace = 2^{7} \cdot 13 \) |
| Artin number field: | Galois closure of 8.0.2303721472.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of \(\Q(\sqrt[4]{52})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 79\cdot 101 + 35\cdot 101^{2} + 9\cdot 101^{3} + 47\cdot 101^{4} + 17\cdot 101^{5} + 45\cdot 101^{6} + 47\cdot 101^{7} + 3\cdot 101^{8} +O(101^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 20\cdot 101 + 53\cdot 101^{2} + 52\cdot 101^{3} + 64\cdot 101^{4} + 26\cdot 101^{5} + 71\cdot 101^{6} + 27\cdot 101^{7} + 3\cdot 101^{8} +O(101^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 + 78\cdot 101 + 24\cdot 101^{2} + 37\cdot 101^{3} + 13\cdot 101^{4} + 67\cdot 101^{5} + 51\cdot 101^{6} + 98\cdot 101^{7} + 48\cdot 101^{8} +O(101^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 + 11\cdot 101 + 57\cdot 101^{2} + 61\cdot 101^{3} + 77\cdot 101^{4} + 98\cdot 101^{5} + 67\cdot 101^{6} + 96\cdot 101^{7} + 64\cdot 101^{8} +O(101^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 79 + 89\cdot 101 + 43\cdot 101^{2} + 39\cdot 101^{3} + 23\cdot 101^{4} + 2\cdot 101^{5} + 33\cdot 101^{6} + 4\cdot 101^{7} + 36\cdot 101^{8} +O(101^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 80 + 22\cdot 101 + 76\cdot 101^{2} + 63\cdot 101^{3} + 87\cdot 101^{4} + 33\cdot 101^{5} + 49\cdot 101^{6} + 2\cdot 101^{7} + 52\cdot 101^{8} +O(101^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 83 + 80\cdot 101 + 47\cdot 101^{2} + 48\cdot 101^{3} + 36\cdot 101^{4} + 74\cdot 101^{5} + 29\cdot 101^{6} + 73\cdot 101^{7} + 97\cdot 101^{8} +O(101^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 93 + 21\cdot 101 + 65\cdot 101^{2} + 91\cdot 101^{3} + 53\cdot 101^{4} + 83\cdot 101^{5} + 55\cdot 101^{6} + 53\cdot 101^{7} + 97\cdot 101^{8} +O(101^{9})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | $0$ |
| $2$ | $4$ | $(2,5,7,4)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(2,4,7,5)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,8,3)(2,7)(4,5)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,3,8,6)(2,7)(4,5)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,4,6,7,8,5,3,2)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,7,3,4,8,2,6,5)$ | $0$ | $0$ |