Basic invariants
Dimension: | $2$ |
Group: | $C_4\wr C_2$ |
Conductor: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
Artin stem field: | Galois closure of 8.0.4917248.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4\wr C_2$ |
Parity: | odd |
Determinant: | 1.112.4t1.b.b |
Projective image: | $D_4$ |
Projective stem field: | Galois closure of 4.2.14336.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 3x^{7} + 6x^{6} - 8x^{5} + 10x^{4} - 9x^{3} + 6x^{2} - 2x + 1 \) . |
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 value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $-2$ |
$2$ | $2$ | $(3,5)(4,7)$ | $0$ |
$4$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $0$ |
$1$ | $4$ | $(1,6,8,2)(3,4,5,7)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,2,8,6)(3,7,5,4)$ | $-2 \zeta_{4}$ |
$2$ | $4$ | $(3,7,5,4)$ | $\zeta_{4} - 1$ |
$2$ | $4$ | $(3,4,5,7)$ | $-\zeta_{4} - 1$ |
$2$ | $4$ | $(1,8)(2,6)(3,4,5,7)$ | $-\zeta_{4} + 1$ |
$2$ | $4$ | $(1,8)(2,6)(3,7,5,4)$ | $\zeta_{4} + 1$ |
$2$ | $4$ | $(1,6,8,2)(3,7,5,4)$ | $0$ |
$4$ | $4$ | $(1,5,8,3)(2,4,6,7)$ | $0$ |
$4$ | $8$ | $(1,4,6,5,8,7,2,3)$ | $0$ |
$4$ | $8$ | $(1,5,2,4,8,3,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.