Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3636\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 101 \) |
| Artin number field: | Galois closure of 8.0.1903751424.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{303})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 112\cdot 157 + 116\cdot 157^{2} + 65\cdot 157^{3} + 67\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 37 + 83\cdot 157 + 55\cdot 157^{2} + 22\cdot 157^{3} + 43\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 + 22\cdot 157 + 49\cdot 157^{2} + 156\cdot 157^{3} + 143\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 69 + 155\cdot 157 + 3\cdot 157^{2} + 119\cdot 157^{3} + 15\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 97 + 124\cdot 157 + 114\cdot 157^{2} + 143\cdot 157^{3} + 39\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 107 + 9\cdot 157 + 120\cdot 157^{2} + 26\cdot 157^{3} + 58\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 123 + 38\cdot 157 + 105\cdot 157^{2} + 88\cdot 157^{3} + 156\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 155 + 81\cdot 157 + 62\cdot 157^{2} + 5\cdot 157^{3} + 103\cdot 157^{4} +O(157^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,5)(3,4)(6,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(4,7)(5,8)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,3,2,6)(4,5,7,8)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,2,3)(4,8,7,5)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,2,4)(3,8,6,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,3,2,6)(4,8,7,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,8,2,5)(3,4,6,7)$ | $0$ | $0$ |