Basic invariants
| Dimension: | $3$ |
| Group: | $A_4$ |
| Conductor: | \(3136\)\(\medspace = 2^{6} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.0.3136.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4$ |
| Parity: | even |
| Projective image: | $A_4$ |
| Projective field: | Galois closure of 4.0.3136.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 + 157\cdot 167 + 9\cdot 167^{2} + 155\cdot 167^{3} + 152\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 38 + 67\cdot 167 + 16\cdot 167^{2} + 108\cdot 167^{3} + 130\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 114 + 25\cdot 167 + 152\cdot 167^{2} + 26\cdot 167^{3} + 5\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 158 + 83\cdot 167 + 155\cdot 167^{2} + 43\cdot 167^{3} + 45\cdot 167^{4} +O(167^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)$ | $0$ |
| $4$ | $3$ | $(1,3,2)$ | $0$ |