Basic invariants
| Dimension: | $3$ |
| Group: | $A_4$ |
| Conductor: | \(190096\)\(\medspace = 2^{4} \cdot 109^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.190096.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.190096.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 5x^{2} + 4x + 5 \)
|
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 6\cdot 43 + 30\cdot 43^{2} + 38\cdot 43^{3} + 42\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 24\cdot 43 + 18\cdot 43^{2} + 5\cdot 43^{3} + 29\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 24\cdot 43 + 12\cdot 43^{2} + 36\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 + 30\cdot 43 + 24\cdot 43^{2} + 41\cdot 43^{3} + 20\cdot 43^{4} +O(43^{5})\)
|
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 value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)$ | $0$ |
| $4$ | $3$ | $(1,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.