Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.8.5120000000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Determinant: | 1.40.4t1.a.b |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 10x^{6} + 25x^{4} - 20x^{2} + 5 \)
|
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 + 178\cdot 199 + 91\cdot 199^{2} + 76\cdot 199^{4} + 116\cdot 199^{5} +O(199^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 85\cdot 199 + 99\cdot 199^{2} + 166\cdot 199^{3} + 117\cdot 199^{4} + 65\cdot 199^{5} +O(199^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 + 26\cdot 199 + 109\cdot 199^{2} + 169\cdot 199^{3} + 196\cdot 199^{4} + 103\cdot 199^{5} +O(199^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 53 + 188\cdot 199 + 177\cdot 199^{2} + 126\cdot 199^{3} + 130\cdot 199^{4} + 158\cdot 199^{5} +O(199^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 146 + 10\cdot 199 + 21\cdot 199^{2} + 72\cdot 199^{3} + 68\cdot 199^{4} + 40\cdot 199^{5} +O(199^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 156 + 172\cdot 199 + 89\cdot 199^{2} + 29\cdot 199^{3} + 2\cdot 199^{4} + 95\cdot 199^{5} +O(199^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 168 + 113\cdot 199 + 99\cdot 199^{2} + 32\cdot 199^{3} + 81\cdot 199^{4} + 133\cdot 199^{5} +O(199^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 180 + 20\cdot 199 + 107\cdot 199^{2} + 198\cdot 199^{3} + 122\cdot 199^{4} + 82\cdot 199^{5} +O(199^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $0$ |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $0$ |
| $2$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $0$ |
| $2$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.