Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(605\)\(\medspace = 5 \cdot 11^{2} \) |
| Artin stem field: | Galois closure of 8.0.221445125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.5.4t1.a.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.1375.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 7x^{6} - 15x^{5} + 18x^{4} - 16x^{3} + 13x^{2} - 5x + 3 \)
|
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 45\cdot 71 + 20\cdot 71^{2} + 41\cdot 71^{3} + 39\cdot 71^{4} + 39\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 41\cdot 71 + 48\cdot 71^{2} + 50\cdot 71^{3} + 2\cdot 71^{4} + 46\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 + 34\cdot 71 + 44\cdot 71^{2} + 59\cdot 71^{3} + 44\cdot 71^{4} + 21\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 + 11\cdot 71 + 40\cdot 71^{2} + 12\cdot 71^{3} + 31\cdot 71^{4} + 41\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 35 + 42\cdot 71 + 25\cdot 71^{2} + 15\cdot 71^{3} + 18\cdot 71^{4} + 9\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 + 16\cdot 71 + 5\cdot 71^{2} + 53\cdot 71^{3} + 39\cdot 71^{4} + 31\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 62 + 43\cdot 71 + 58\cdot 71^{2} + 42\cdot 71^{3} + 29\cdot 71^{4} + 54\cdot 71^{5} +O(71^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 63 + 48\cdot 71 + 40\cdot 71^{2} + 8\cdot 71^{3} + 7\cdot 71^{4} + 40\cdot 71^{5} +O(71^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $-2$ |
| $2$ | $2$ | $(3,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,3)(4,6)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,6,2,7)(3,5,8,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,2,6)(3,4,8,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,2,6)(3,5,8,4)$ | $0$ |
| $2$ | $4$ | $(3,4,8,5)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(3,5,8,4)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,2)(3,5,8,4)(6,7)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,2)(3,4,8,5)(6,7)$ | $-\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,4,2,5)(3,7,8,6)$ | $0$ |
| $4$ | $8$ | $(1,8,6,4,2,3,7,5)$ | $0$ |
| $4$ | $8$ | $(1,4,7,8,2,5,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.