Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(1012971\)\(\medspace = 3 \cdot 59^{2} \cdot 97 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.3038913.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.291.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{97})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 281x^{2} - 74x + 21052 \)
|
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 31 + 29\cdot 31^{3} + 22\cdot 31^{4} + 18\cdot 31^{5} + 26\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 11\cdot 31 + 20\cdot 31^{2} + 17\cdot 31^{3} + 15\cdot 31^{4} + 13\cdot 31^{5} + 6\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 5\cdot 31 + 17\cdot 31^{2} + 4\cdot 31^{3} + 28\cdot 31^{4} + 18\cdot 31^{5} + 27\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 + 12\cdot 31 + 24\cdot 31^{2} + 10\cdot 31^{3} + 26\cdot 31^{4} + 10\cdot 31^{5} + 31^{6} +O(31^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(2,3)$ | $0$ |
| $2$ | $4$ | $(1,2,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.