Basic invariants
| Dimension: | $3$ |
| Group: | $A_4$ |
| Conductor: | \(132496\)\(\medspace = 2^{4} \cdot 7^{2} \cdot 13^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.0.132496.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4$ |
| Parity: | even |
| Projective image: | $A_4$ |
| Projective field: | Galois closure of 4.0.132496.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 72\cdot 83 + 51\cdot 83^{2} + 40\cdot 83^{3} + 22\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 22\cdot 83 + 30\cdot 83^{2} + 34\cdot 83^{3} + 39\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 61 + 47\cdot 83 + 41\cdot 83^{2} + 18\cdot 83^{3} + 45\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 77 + 23\cdot 83 + 42\cdot 83^{2} + 72\cdot 83^{3} + 58\cdot 83^{4} +O(83^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)$ | $0$ |
| $4$ | $3$ | $(1,3,2)$ | $0$ |