Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Artin number field: | Galois closure of 8.0.260112384.11 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-14})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 349 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 69 + 230\cdot 349 + 162\cdot 349^{2} + 323\cdot 349^{3} + 215\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 110 + 116\cdot 349 + 323\cdot 349^{2} + 114\cdot 349^{3} + 243\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 148 + 315\cdot 349 + 273\cdot 349^{2} + 163\cdot 349^{3} + 278\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 154 + 117\cdot 349 + 12\cdot 349^{2} + 135\cdot 349^{3} + 173\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 195 + 231\cdot 349 + 336\cdot 349^{2} + 213\cdot 349^{3} + 175\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 201 + 33\cdot 349 + 75\cdot 349^{2} + 185\cdot 349^{3} + 70\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 239 + 232\cdot 349 + 25\cdot 349^{2} + 234\cdot 349^{3} + 105\cdot 349^{4} +O(349^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 280 + 118\cdot 349 + 186\cdot 349^{2} + 25\cdot 349^{3} + 133\cdot 349^{4} +O(349^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ | $0$ |