Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(4645\)\(\medspace = 5 \cdot 929 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.4315205.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | even |
Determinant: | 1.4645.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{929})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 2x^{3} + 32x^{2} - 31x + 8 \) . |
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ |
\( 7 + 26\cdot 29 + 10\cdot 29^{2} + 22\cdot 29^{3} + 27\cdot 29^{4} + 24\cdot 29^{5} +O(29^{6})\)
$r_{ 2 }$ |
$=$ |
\( 9 + 2\cdot 29 + 13\cdot 29^{2} + 2\cdot 29^{3} + 15\cdot 29^{4} + 4\cdot 29^{5} +O(29^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 21 + 26\cdot 29 + 15\cdot 29^{2} + 26\cdot 29^{3} + 13\cdot 29^{4} + 24\cdot 29^{5} +O(29^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 23 + 2\cdot 29 + 18\cdot 29^{2} + 6\cdot 29^{3} + 29^{4} + 4\cdot 29^{5} +O(29^{6})\)
| |
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.