Basic invariants
Dimension: | $3$ |
Group: | $A_4$ |
Conductor: | \(205209\)\(\medspace = 3^{2} \cdot 151^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.205209.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $A_4$ |
Projective stem field: | Galois closure of 4.0.205209.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 4x^{2} - 3x + 12 \) . |
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 6 + 17\cdot 29 + 6\cdot 29^{2} + 11\cdot 29^{3} + 16\cdot 29^{4} +O(29^{5})\)
$r_{ 2 }$ |
$=$ |
\( 8 + 14\cdot 29 + 27\cdot 29^{2} + 21\cdot 29^{3} + 24\cdot 29^{4} +O(29^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 18 + 14\cdot 29 + 6\cdot 29^{2} + 9\cdot 29^{3} + 23\cdot 29^{4} +O(29^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 27 + 11\cdot 29 + 17\cdot 29^{2} + 15\cdot 29^{3} + 22\cdot 29^{4} +O(29^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$4$ | $3$ | $(1,2,3)$ | $0$ |
$4$ | $3$ | $(1,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.