Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(544\)\(\medspace = 2^{5} \cdot 17 \) |
| Artin stem field: | Galois closure of 8.0.321978368.6 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.136.4t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.39304.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 6x^{6} - 6x^{4} + 4x^{3} + 2x^{2} + 9 \)
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The roots of $f$ are computed in $\Q_{ 577 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 62 + 295\cdot 577 + 138\cdot 577^{2} + 28\cdot 577^{3} + 561\cdot 577^{4} +O(577^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 284 + 218\cdot 577 + 538\cdot 577^{2} + 155\cdot 577^{3} + 10\cdot 577^{4} +O(577^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 291 + 463\cdot 577 + 94\cdot 577^{2} + 385\cdot 577^{3} + 313\cdot 577^{4} +O(577^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 375 + 138\cdot 577 + 285\cdot 577^{2} + 568\cdot 577^{3} + 39\cdot 577^{4} +O(577^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 433 + 520\cdot 577 + 575\cdot 577^{2} + 548\cdot 577^{3} + 502\cdot 577^{4} +O(577^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 438 + 559\cdot 577 + 329\cdot 577^{2} + 261\cdot 577^{3} + 347\cdot 577^{4} +O(577^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 481 + 487\cdot 577 + 359\cdot 577^{2} + 350\cdot 577^{3} + 176\cdot 577^{4} +O(577^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 525 + 200\cdot 577 + 562\cdot 577^{2} + 8\cdot 577^{3} + 356\cdot 577^{4} +O(577^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,6)(3,7)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,6)(3,7)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,8)(3,4)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,3,6,7)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,7,6,3)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(2,7,6,3)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(2,3,6,7)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,3,6,7)(4,5)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,7,6,3)(4,5)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,5,8,4)(2,7,6,3)$ | $0$ |
| $4$ | $4$ | $(1,3,8,7)(2,5,6,4)$ | $0$ |
| $4$ | $8$ | $(1,2,5,3,8,6,4,7)$ | $0$ |
| $4$ | $8$ | $(1,3,4,2,8,7,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.