Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(825\)\(\medspace = 3 \cdot 5^{2} \cdot 11 \) |
| Artin stem field: | Galois closure of 8.0.16471125.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.165.4t1.a.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.12375.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + x^{6} + 2x^{5} - 3x^{4} - 2x^{3} + x^{2} + x + 1 \)
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The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 52 + 39\cdot 269 + 148\cdot 269^{2} + 164\cdot 269^{3} + 60\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 60 + 178\cdot 269 + 199\cdot 269^{2} + 152\cdot 269^{3} + 180\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 70 + 90\cdot 269 + 188\cdot 269^{2} + 123\cdot 269^{3} + 265\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 73 + 25\cdot 269 + 98\cdot 269^{2} + 200\cdot 269^{3} + 195\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 128 + 88\cdot 269 + 56\cdot 269^{2} + 41\cdot 269^{3} + 179\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 130 + 246\cdot 269 + 231\cdot 269^{2} + 258\cdot 269^{3} + 263\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 145 + 70\cdot 269 + 16\cdot 269^{2} + 109\cdot 269^{3} + 186\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 150 + 68\cdot 269 + 137\cdot 269^{2} + 25\cdot 269^{3} + 13\cdot 269^{4} +O(269^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,6)(3,4)(5,7)$ | $-2$ |
| $2$ | $2$ | $(2,6)(3,4)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,7)(4,8)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,7,8,5)(2,4,6,3)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,8,7)(2,3,6,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(2,4,6,3)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(2,3,6,4)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,3,6,4)(5,7)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,4,6,3)(5,7)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,5,8,7)(2,4,6,3)$ | $0$ |
| $4$ | $4$ | $(1,4,8,3)(2,7,6,5)$ | $0$ |
| $4$ | $8$ | $(1,2,7,4,8,6,5,3)$ | $0$ |
| $4$ | $8$ | $(1,4,5,2,8,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.