Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(275\)\(\medspace = 5^{2} \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.228765625.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.11.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{6} + 16x^{4} - x^{2} + 1 \)
|
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 6\cdot 71 + 37\cdot 71^{2} + 28\cdot 71^{3} + 22\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 47\cdot 71 + 32\cdot 71^{2} + 46\cdot 71^{3} + 8\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 + 13\cdot 71 + 45\cdot 71^{2} + 64\cdot 71^{3} + 25\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 + 37\cdot 71 + 3\cdot 71^{2} + 45\cdot 71^{3} + 41\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 44 + 33\cdot 71 + 67\cdot 71^{2} + 25\cdot 71^{3} + 29\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 50 + 57\cdot 71 + 25\cdot 71^{2} + 6\cdot 71^{3} + 45\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 59 + 23\cdot 71 + 38\cdot 71^{2} + 24\cdot 71^{3} + 62\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 65 + 64\cdot 71 + 33\cdot 71^{2} + 42\cdot 71^{3} + 48\cdot 71^{4} +O(71^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,7,4)(2,5,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.