Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(448\)\(\medspace = 2^{6} \cdot 7 \) |
| Artin number field: | Galois closure of 8.0.78675968.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.14336.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 18\cdot 127 + 121\cdot 127^{2} + 22\cdot 127^{3} + 60\cdot 127^{4} + 70\cdot 127^{5} + 26\cdot 127^{6} +O(127^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 116\cdot 127 + 49\cdot 127^{2} + 123\cdot 127^{3} + 65\cdot 127^{4} + 93\cdot 127^{5} + 91\cdot 127^{6} +O(127^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 34 + 110\cdot 127 + 28\cdot 127^{2} + 59\cdot 127^{3} + 111\cdot 127^{4} + 47\cdot 127^{5} + 121\cdot 127^{6} +O(127^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 51 + 17\cdot 127 + 8\cdot 127^{2} + 24\cdot 127^{3} + 123\cdot 127^{4} + 64\cdot 127^{5} + 83\cdot 127^{6} +O(127^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 76 + 109\cdot 127 + 118\cdot 127^{2} + 102\cdot 127^{3} + 3\cdot 127^{4} + 62\cdot 127^{5} + 43\cdot 127^{6} +O(127^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 93 + 16\cdot 127 + 98\cdot 127^{2} + 67\cdot 127^{3} + 15\cdot 127^{4} + 79\cdot 127^{5} + 5\cdot 127^{6} +O(127^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 95 + 10\cdot 127 + 77\cdot 127^{2} + 3\cdot 127^{3} + 61\cdot 127^{4} + 33\cdot 127^{5} + 35\cdot 127^{6} +O(127^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 114 + 108\cdot 127 + 5\cdot 127^{2} + 104\cdot 127^{3} + 66\cdot 127^{4} + 56\cdot 127^{5} + 100\cdot 127^{6} +O(127^{7})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,8,5)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,5,8,4)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,5,8,4)(2,7)(3,6)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,4,8,5)(2,7)(3,6)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,3,5,2,8,6,4,7)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,2,4,3,8,7,5,6)$ | $0$ | $0$ |