Basic invariants
| Dimension: | $3$ |
| Group: | $A_4$ |
| Conductor: | \(247009\)\(\medspace = 7^{2} \cdot 71^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.0.247009.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4$ |
| Parity: | even |
| Projective image: | $A_4$ |
| Projective field: | Galois closure of 4.0.247009.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 7\cdot 13 + 5\cdot 13^{2} + 7\cdot 13^{3} + 4\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 5\cdot 13 + 6\cdot 13^{2} + 8\cdot 13^{3} +O(13^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 + 5\cdot 13 + 4\cdot 13^{2} + 13^{3} + 12\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 + 7\cdot 13 + 9\cdot 13^{2} + 8\cdot 13^{3} + 8\cdot 13^{4} +O(13^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)$ | $0$ |
| $4$ | $3$ | $(1,3,2)$ | $0$ |