Basic invariants
Dimension: | $3$ |
Group: | $A_4$ |
Conductor: | \(227529\)\(\medspace = 3^{4} \cdot 53^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.227529.1 |
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 4.0.227529.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 3x^{2} - 8x + 11 \) . |
The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 3 + 5\cdot 17 + 8\cdot 17^{2} + 7\cdot 17^{3} +O(17^{5})\)
$r_{ 2 }$ |
$=$ |
\( 8 + 4\cdot 17 + 12\cdot 17^{2} + 14\cdot 17^{4} +O(17^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 10 + 10\cdot 17 + 17^{2} + 11\cdot 17^{3} + 6\cdot 17^{4} +O(17^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 14 + 13\cdot 17 + 11\cdot 17^{2} + 14\cdot 17^{3} + 12\cdot 17^{4} +O(17^{5})\)
| |
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.