Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(667\)\(\medspace = 23 \cdot 29 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.15341.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.667.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-23}, \sqrt{29})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 2x^{2} + 4x + 16 \)
|
The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 104\cdot 167 + 166\cdot 167^{2} + 141\cdot 167^{3} + 78\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 + 115\cdot 167 + 86\cdot 167^{2} + 119\cdot 167^{3} + 35\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 50 + 48\cdot 167 + 79\cdot 167^{2} + 116\cdot 167^{3} + 129\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 71 + 66\cdot 167 + 167^{2} + 123\cdot 167^{3} + 89\cdot 167^{4} +O(167^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.