Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
| Artin number field: | Galois closure of 8.0.4917248.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.14336.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 239 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 30 + 93\cdot 239 + 155\cdot 239^{2} + 152\cdot 239^{3} + 154\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 92 + 154\cdot 239 + 26\cdot 239^{2} + 233\cdot 239^{3} + 21\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 122 + 203\cdot 239 + 211\cdot 239^{2} + 50\cdot 239^{3} + 166\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 126 + 68\cdot 239 + 25\cdot 239^{2} + 86\cdot 239^{3} + 60\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 187 + 73\cdot 239 + 178\cdot 239^{2} + 232\cdot 239^{3} + 208\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 194 + 202\cdot 239 + 87\cdot 239^{2} + 65\cdot 239^{3} + 69\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 211 + 85\cdot 239 + 207\cdot 239^{2} + 116\cdot 239^{3} + 137\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 236 + 73\cdot 239 + 63\cdot 239^{2} + 18\cdot 239^{3} + 137\cdot 239^{4} +O(239^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(3,5)(4,7)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,6,8,2)(3,4,5,7)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,6)(3,7,5,4)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(3,7,5,4)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(3,4,5,7)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,6)(3,4,5,7)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,6)(3,7,5,4)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,6,8,2)(3,7,5,4)$ | $0$ | $0$ |
| $4$ | $4$ | $(1,5,8,3)(2,4,6,7)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,4,6,5,8,7,2,3)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,5,2,4,8,3,6,7)$ | $0$ | $0$ |