Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1044\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 29 \) |
| Artin stem field: | Galois closure of 8.0.156950784.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.116.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{87})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 11x^{6} - 8x^{5} + 31x^{4} + 2x^{3} + 29x^{2} + 20x + 13 \)
|
The roots of $f$ are computed in $\Q_{ 277 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 74 + 9\cdot 277 + 18\cdot 277^{2} + 222\cdot 277^{3} + 192\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 76 + 156\cdot 277 + 30\cdot 277^{2} + 84\cdot 277^{3} + 74\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 101 + 254\cdot 277 + 11\cdot 277^{2} + 195\cdot 277^{3} + 86\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 120 + 107\cdot 277 + 151\cdot 277^{2} + 69\cdot 277^{3} + 74\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 227 + 186\cdot 277 + 33\cdot 277^{2} + 42\cdot 277^{3} + 143\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 258 + 35\cdot 277 + 83\cdot 277^{2} + 205\cdot 277^{3} + 41\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 260 + 174\cdot 277 + 172\cdot 277^{2} + 49\cdot 277^{3} + 175\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 271 + 182\cdot 277 + 52\cdot 277^{2} + 240\cdot 277^{3} + 42\cdot 277^{4} +O(277^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,8)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,4,7,6)(2,5,3,8)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,7,4)(2,8,3,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,7,3)(4,5,6,8)$ | $0$ |
| $2$ | $4$ | $(1,8,7,5)(2,6,3,4)$ | $0$ |
| $2$ | $4$ | $(1,6,7,4)(2,5,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.