Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(3528\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.34153975296.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.0.2744.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 443 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 54 + 127\cdot 443 + 372\cdot 443^{2} + 145\cdot 443^{3} + 201\cdot 443^{4} +O(443^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 275 + 30\cdot 443 + 97\cdot 443^{2} + 110\cdot 443^{3} + 31\cdot 443^{4} +O(443^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 325 + 275\cdot 443 + 28\cdot 443^{2} + 437\cdot 443^{3} + 328\cdot 443^{4} +O(443^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 373 + 235\cdot 443 + 406\cdot 443^{2} + 188\cdot 443^{3} + 381\cdot 443^{4} +O(443^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 390 + 351\cdot 443 + 168\cdot 443^{2} + 369\cdot 443^{3} + 435\cdot 443^{4} +O(443^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 404 + 268\cdot 443 + 78\cdot 443^{2} + 403\cdot 443^{3} + 399\cdot 443^{4} +O(443^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 416 + 4\cdot 443 + 414\cdot 443^{2} + 435\cdot 443^{3} + 320\cdot 443^{4} +O(443^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 424 + 33\cdot 443 + 206\cdot 443^{2} + 124\cdot 443^{3} + 115\cdot 443^{4} +O(443^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $0$ | $0$ |
| $4$ | $2$ | $(2,7)(4,8)(5,6)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,6,3,5)(2,8,4,7)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,8,5,2,3,7,6,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,2,6,8,3,4,5,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |