Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(723\)\(\medspace = 3 \cdot 241 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.0.2169.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{241})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 23 + 114\cdot 181 + 171\cdot 181^{2} + 75\cdot 181^{3} + 50\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 110 + 68\cdot 181 + 93\cdot 181^{2} + 108\cdot 181^{3} + 20\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 111 + 123\cdot 181 + 161\cdot 181^{2} + 116\cdot 181^{3} + 127\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 119 + 55\cdot 181 + 116\cdot 181^{2} + 60\cdot 181^{3} + 163\cdot 181^{4} +O(181^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $2$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $4$ | $(1,4,2,3)$ | $0$ |