Basic invariants
Dimension: | $2$ |
Group: | $C_4\wr C_2$ |
Conductor: | \(165\)\(\medspace = 3 \cdot 5 \cdot 11 \) |
Artin number field: | Galois closure of 8.0.16471125.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4\wr C_2$ |
Parity: | odd |
Projective image: | $D_4$ |
Projective field: | Galois closure of 4.2.12375.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 52 + 39\cdot 269 + 148\cdot 269^{2} + 164\cdot 269^{3} + 60\cdot 269^{4} +O(269^{5})\)
$r_{ 2 }$ |
$=$ |
\( 60 + 178\cdot 269 + 199\cdot 269^{2} + 152\cdot 269^{3} + 180\cdot 269^{4} +O(269^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 70 + 90\cdot 269 + 188\cdot 269^{2} + 123\cdot 269^{3} + 265\cdot 269^{4} +O(269^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 73 + 25\cdot 269 + 98\cdot 269^{2} + 200\cdot 269^{3} + 195\cdot 269^{4} +O(269^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 128 + 88\cdot 269 + 56\cdot 269^{2} + 41\cdot 269^{3} + 179\cdot 269^{4} +O(269^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 130 + 246\cdot 269 + 231\cdot 269^{2} + 258\cdot 269^{3} + 263\cdot 269^{4} +O(269^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 145 + 70\cdot 269 + 16\cdot 269^{2} + 109\cdot 269^{3} + 186\cdot 269^{4} +O(269^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 150 + 68\cdot 269 + 137\cdot 269^{2} + 25\cdot 269^{3} + 13\cdot 269^{4} +O(269^{5})\)
| |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$1$ | $2$ | $(1,8)(2,6)(3,4)(5,7)$ | $-2$ | $-2$ |
$2$ | $2$ | $(2,6)(3,4)$ | $0$ | $0$ |
$4$ | $2$ | $(1,3)(2,7)(4,8)(5,6)$ | $0$ | $0$ |
$1$ | $4$ | $(1,7,8,5)(2,4,6,3)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
$1$ | $4$ | $(1,5,8,7)(2,3,6,4)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
$2$ | $4$ | $(2,4,6,3)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
$2$ | $4$ | $(2,3,6,4)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
$2$ | $4$ | $(1,8)(2,3,6,4)(5,7)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
$2$ | $4$ | $(1,8)(2,4,6,3)(5,7)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
$2$ | $4$ | $(1,5,8,7)(2,4,6,3)$ | $0$ | $0$ |
$4$ | $4$ | $(1,4,8,3)(2,7,6,5)$ | $0$ | $0$ |
$4$ | $8$ | $(1,2,7,4,8,6,5,3)$ | $0$ | $0$ |
$4$ | $8$ | $(1,4,5,2,8,3,7,6)$ | $0$ | $0$ |