Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(1024\)\(\medspace = 2^{10} \) |
| Artin stem field: | Galois closure of 8.4.2147483648.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.16.4t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\zeta_{8})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 8x^{6} - 12x^{4} + 2 \)
|
The roots of $f$ are computed in $\Q_{ 241 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 28\cdot 241 + 137\cdot 241^{2} + 157\cdot 241^{3} + 222\cdot 241^{4} + 164\cdot 241^{5} + 231\cdot 241^{6} +O(241^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 215\cdot 241 + 191\cdot 241^{2} + 140\cdot 241^{3} + 193\cdot 241^{4} + 186\cdot 241^{5} + 71\cdot 241^{6} +O(241^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 40 + 134\cdot 241 + 12\cdot 241^{2} + 143\cdot 241^{3} + 38\cdot 241^{4} + 113\cdot 241^{5} + 15\cdot 241^{6} +O(241^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 56 + 25\cdot 241 + 27\cdot 241^{2} + 133\cdot 241^{3} + 97\cdot 241^{4} + 70\cdot 241^{5} + 223\cdot 241^{6} +O(241^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 185 + 215\cdot 241 + 213\cdot 241^{2} + 107\cdot 241^{3} + 143\cdot 241^{4} + 170\cdot 241^{5} + 17\cdot 241^{6} +O(241^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 201 + 106\cdot 241 + 228\cdot 241^{2} + 97\cdot 241^{3} + 202\cdot 241^{4} + 127\cdot 241^{5} + 225\cdot 241^{6} +O(241^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 223 + 25\cdot 241 + 49\cdot 241^{2} + 100\cdot 241^{3} + 47\cdot 241^{4} + 54\cdot 241^{5} + 169\cdot 241^{6} +O(241^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 238 + 212\cdot 241 + 103\cdot 241^{2} + 83\cdot 241^{3} + 18\cdot 241^{4} + 76\cdot 241^{5} + 9\cdot 241^{6} +O(241^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,4,2,3,8,5,7,6)$ | $0$ |
| $2$ | $8$ | $(1,3,7,4,8,6,2,5)$ | $0$ |
| $2$ | $8$ | $(1,3,2,5,8,6,7,4)$ | $0$ |
| $2$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.