Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Artin number field: | Galois closure of 8.0.2401000000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 38\cdot 79 + 77\cdot 79^{2} + 17\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 9\cdot 79 + 71\cdot 79^{2} + 75\cdot 79^{3} + 77\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 53\cdot 79 + 35\cdot 79^{2} + 6\cdot 79^{3} + 12\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 + 67\cdot 79 + 27\cdot 79^{2} + 17\cdot 79^{3} + 7\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 26 + 66\cdot 79 + 26\cdot 79^{2} + 32\cdot 79^{3} + 75\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 28 + 16\cdot 79 + 6\cdot 79^{2} + 62\cdot 79^{3} + 56\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 48 + 2\cdot 79 + 24\cdot 79^{2} + 16\cdot 79^{3} + 17\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 74 + 62\cdot 79 + 46\cdot 79^{2} + 25\cdot 79^{3} + 52\cdot 79^{4} +O(79^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,6)(2,3)(4,8)(5,7)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,4)(3,8)(6,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $0$ | $0$ |
| $2$ | $2$ | $(4,8)(5,7)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,3,6,2)(4,5,8,7)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,6,3)(4,7,8,5)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,6,4)(2,5,3,7)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,3,6,2)(4,7,8,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,5,6,7)(2,4,3,8)$ | $0$ | $0$ |