Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(328\)\(\medspace = 2^{3} \cdot 41 \) |
| Artin number field: | Galois closure of 8.0.282300416.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.551368.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 163 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 37\cdot 163 + 87\cdot 163^{2} + 160\cdot 163^{3} + 143\cdot 163^{4} + 66\cdot 163^{5} + 120\cdot 163^{6} + 29\cdot 163^{7} + 133\cdot 163^{8} + 106\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 35\cdot 163 + 93\cdot 163^{2} + 152\cdot 163^{3} + 113\cdot 163^{4} + 2\cdot 163^{5} + 140\cdot 163^{6} + 143\cdot 163^{7} + 128\cdot 163^{8} + 88\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 45 + 149\cdot 163 + 53\cdot 163^{2} + 10\cdot 163^{3} + 132\cdot 163^{4} + 156\cdot 163^{5} + 116\cdot 163^{6} + 147\cdot 163^{7} + 5\cdot 163^{8} + 114\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 78 + 156\cdot 163 + 160\cdot 163^{2} + 71\cdot 163^{3} + 71\cdot 163^{4} + 126\cdot 163^{5} + 142\cdot 163^{6} + 163^{7} + 50\cdot 163^{8} + 17\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 101 + 153\cdot 163 + 58\cdot 163^{2} + 6\cdot 163^{3} + 126\cdot 163^{4} + 34\cdot 163^{5} + 144\cdot 163^{6} + 84\cdot 163^{7} + 71\cdot 163^{8} + 150\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 114 + 154\cdot 163 + 15\cdot 163^{2} + 79\cdot 163^{3} + 42\cdot 163^{4} + 38\cdot 163^{5} + 49\cdot 163^{6} + 64\cdot 163^{7} + 76\cdot 163^{8} + 160\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 133 + 161\cdot 163 + 95\cdot 163^{2} + 50\cdot 163^{3} + 18\cdot 163^{4} + 65\cdot 163^{5} + 4\cdot 163^{6} + 69\cdot 163^{7} + 39\cdot 163^{8} + 32\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 150 + 129\cdot 163 + 85\cdot 163^{2} + 120\cdot 163^{3} + 3\cdot 163^{4} + 161\cdot 163^{5} + 96\cdot 163^{6} + 110\cdot 163^{7} + 146\cdot 163^{8} + 144\cdot 163^{9} +O(163^{10})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,7)(5,6)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,6,7,5)(2,4,8,3)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,7,6)(2,3,8,4)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,7,5)(2,3,8,4)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,6,7,5)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,5,7,6)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,7)(2,3,8,4)(5,6)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,7)(2,4,8,3)(5,6)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,4,7,3)(2,6,8,5)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,4,6,8,7,3,5,2)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,8,5,4,7,2,6,3)$ | $0$ | $0$ |