Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(3143\)\(\medspace = 7 \cdot 449 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.2.1411207.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-7}, \sqrt{449})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 40 + 71\cdot 127 + 67\cdot 127^{2} + 41\cdot 127^{3} + 99\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 59 + 57\cdot 127 + 60\cdot 127^{2} + 69\cdot 127^{3} + 8\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 68 + 69\cdot 127 + 66\cdot 127^{2} + 57\cdot 127^{3} + 118\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 87 + 55\cdot 127 + 59\cdot 127^{2} + 85\cdot 127^{3} + 27\cdot 127^{4} +O(127^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |