Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(975\)\(\medspace = 3 \cdot 5^{2} \cdot 13 \) |
| Artin stem field: | Galois closure of 8.0.213890625.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.195.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{13})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 10x^{6} - 9x^{5} + 34x^{4} - 29x^{3} + 45x^{2} - 36x + 16 \)
|
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 + 50\cdot 79 + 64\cdot 79^{2} + 47\cdot 79^{3} + 58\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 19\cdot 79 + 24\cdot 79^{2} + 74\cdot 79^{3} + 45\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 + 49\cdot 79 + 27\cdot 79^{2} + 25\cdot 79^{3} + 50\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 48 + 30\cdot 79 + 3\cdot 79^{2} + 60\cdot 79^{3} + 8\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 57 + 76\cdot 79 + 29\cdot 79^{2} + 75\cdot 79^{3} +O(79^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 66 + 35\cdot 79 + 41\cdot 79^{2} + 43\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 70 + 38\cdot 79 + 67\cdot 79^{2} + 67\cdot 79^{3} + 23\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 74 + 14\cdot 79 + 57\cdot 79^{2} + 38\cdot 79^{4} +O(79^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(2,8)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $1$ | $4$ | $(1,5,7,3)(2,4,8,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,7,5)(2,6,8,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,7,8)(3,6,5,4)$ | $0$ |
| $2$ | $4$ | $(1,5,7,3)(2,6,8,4)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,5,8,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.