Basic invariants
| Dimension: | $3$ |
| Group: | $A_4$ |
| Conductor: | \(219024\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 13^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.0.219024.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4$ |
| Parity: | even |
| Projective image: | $A_4$ |
| Projective field: | Galois closure of 4.0.219024.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 49\cdot 53 + 42\cdot 53^{2} + 4\cdot 53^{3} + 15\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 + 51\cdot 53 + 15\cdot 53^{2} + 12\cdot 53^{3} + 27\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 + 12\cdot 53 + 34\cdot 53^{3} + 9\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 40 + 45\cdot 53 + 46\cdot 53^{2} + 53^{3} + 53^{4} +O(53^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)$ | $0$ |
| $4$ | $3$ | $(1,3,2)$ | $0$ |