Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(399\)\(\medspace = 3 \cdot 7 \cdot 19 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.1197.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.399.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{133})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 5x^{2} + 3x + 9 \) . |
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 7 + 84\cdot 97 + 69\cdot 97^{2} + 23\cdot 97^{3} + 14\cdot 97^{4} +O(97^{5})\)
$r_{ 2 }$ |
$=$ |
\( 12 + 72\cdot 97 + 71\cdot 97^{2} + 87\cdot 97^{3} + 46\cdot 97^{4} +O(97^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 24 + 74\cdot 97 + 18\cdot 97^{2} + 58\cdot 97^{3} + 81\cdot 97^{4} +O(97^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 55 + 60\cdot 97 + 33\cdot 97^{2} + 24\cdot 97^{3} + 51\cdot 97^{4} +O(97^{5})\)
| |
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.