Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.1008.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.84.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{7})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 5x^{2} + 7 \) . |
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 11 + 2\cdot 31 + 24\cdot 31^{2} + 17\cdot 31^{3} + 3\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$ |
$=$ |
\( 15 + 24\cdot 31 + 17\cdot 31^{2} + 6\cdot 31^{3} + 3\cdot 31^{4} +O(31^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 16 + 6\cdot 31 + 13\cdot 31^{2} + 24\cdot 31^{3} + 27\cdot 31^{4} +O(31^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 20 + 28\cdot 31 + 6\cdot 31^{2} + 13\cdot 31^{3} + 27\cdot 31^{4} +O(31^{5})\)
| |
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.