Basic invariants
| Dimension: | $3$ |
| Group: | $A_4$ |
| Conductor: | \(17689\)\(\medspace = 7^{2} \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.0.17689.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4$ |
| Parity: | even |
| Projective image: | $A_4$ |
| Projective field: | Galois closure of 4.0.17689.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 109 + 18\cdot 163 + 38\cdot 163^{2} + 114\cdot 163^{3} + 15\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 117 + 87\cdot 163 + 157\cdot 163^{2} + 15\cdot 163^{3} + 62\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 119 + 122\cdot 163 + 70\cdot 163^{2} + 88\cdot 163^{3} + 108\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 144 + 96\cdot 163 + 59\cdot 163^{2} + 107\cdot 163^{3} + 139\cdot 163^{4} +O(163^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)$ | $0$ |
| $4$ | $3$ | $(1,3,2)$ | $0$ |