Basic invariants
| Dimension: | $3$ |
| Group: | $A_4$ |
| Conductor: | \(162409\)\(\medspace = 13^{2} \cdot 31^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.162409.1 |
| 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 4.0.162409.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 6x^{2} + 7x + 18 \)
|
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 30 + 31\cdot 53 + 8\cdot 53^{2} + 7\cdot 53^{3} + 14\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 35 + 51\cdot 53 + 41\cdot 53^{2} + 4\cdot 53^{3} + 14\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 47 + 49\cdot 53 + 21\cdot 53^{2} + 9\cdot 53^{3} + 2\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 48 + 25\cdot 53 + 33\cdot 53^{2} + 31\cdot 53^{3} + 22\cdot 53^{4} +O(53^{5})\)
|
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.