Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(6975\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 31 \) |
| Artin stem field: | Galois closure of 8.8.54731953125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Determinant: | 1.155.4t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{93})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} - 17x^{6} + 39x^{5} + 105x^{4} - 126x^{3} - 242x^{2} + 27x + 61 \)
|
The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 86\cdot 269 + 126\cdot 269^{2} + 86\cdot 269^{3} + 113\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 45\cdot 269 + 147\cdot 269^{2} + 169\cdot 269^{3} + 210\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 88 + 143\cdot 269 + 113\cdot 269^{2} + 253\cdot 269^{3} + 189\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 159 + 153\cdot 269 + 192\cdot 269^{2} + 200\cdot 269^{3} + 155\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 176 + 7\cdot 269 + 85\cdot 269^{2} + 110\cdot 269^{3} + 125\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 188 + 144\cdot 269 + 110\cdot 269^{2} + 177\cdot 269^{3} + 174\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 210 + 264\cdot 269 + 30\cdot 269^{2} + 135\cdot 269^{3} + 151\cdot 269^{4} +O(269^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 216 + 230\cdot 269 + 212\cdot 269^{3} + 223\cdot 269^{4} +O(269^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,8)$ | $0$ |
| $1$ | $4$ | $(1,3,5,6)(2,8,7,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,5,3)(2,4,7,8)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,5,6)(2,4,7,8)$ | $0$ |
| $2$ | $8$ | $(1,4,3,2,5,8,6,7)$ | $0$ |
| $2$ | $8$ | $(1,2,6,4,5,7,3,8)$ | $0$ |
| $2$ | $8$ | $(1,4,6,7,5,8,3,2)$ | $0$ |
| $2$ | $8$ | $(1,7,3,4,5,2,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.