Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(1225\)\(\medspace = 5^{2} \cdot 7^{2} \) |
| Artin stem field: | Galois closure of 8.4.9191328125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.5.4t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-7})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 13x^{6} - 13x^{5} + 25x^{4} + 38x^{3} - 33x^{2} - 34x + 11 \)
|
The roots of $f$ are computed in $\Q_{ 281 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 59 + 20\cdot 281 + 4\cdot 281^{2} + 109\cdot 281^{3} + 26\cdot 281^{4} +O(281^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 94 + 154\cdot 281 + 133\cdot 281^{2} + 251\cdot 281^{3} + 160\cdot 281^{4} +O(281^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 125 + 163\cdot 281 + 152\cdot 281^{2} + 22\cdot 281^{3} + 108\cdot 281^{4} +O(281^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 142 + 113\cdot 281 + 139\cdot 281^{2} + 240\cdot 281^{3} + 199\cdot 281^{4} +O(281^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 151 + 253\cdot 281 + 31\cdot 281^{2} + 135\cdot 281^{3} + 244\cdot 281^{4} +O(281^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 152 + 105\cdot 281 + 86\cdot 281^{2} + 4\cdot 281^{3} + 177\cdot 281^{4} +O(281^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 172 + 245\cdot 281 + 234\cdot 281^{2} + 212\cdot 281^{3} + 125\cdot 281^{4} +O(281^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 230 + 67\cdot 281 + 60\cdot 281^{2} + 148\cdot 281^{3} + 81\cdot 281^{4} +O(281^{5})\)
|
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 value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $-2$ |
| $2$ | $2$ | $(3,7)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,2,8,4)(3,5,7,6)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,2)(3,6,7,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,8,4)(3,6,7,5)$ | $0$ |
| $2$ | $8$ | $(1,6,2,3,8,5,4,7)$ | $0$ |
| $2$ | $8$ | $(1,3,4,6,8,7,2,5)$ | $0$ |
| $2$ | $8$ | $(1,6,4,7,8,5,2,3)$ | $0$ |
| $2$ | $8$ | $(1,7,2,6,8,3,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.