Basic invariants
| Dimension: | $3$ |
| Group: | $A_4$ |
| Conductor: | \(29241\)\(\medspace = 3^{4} \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.29241.1 |
| 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 4.0.29241.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 3x^{2} + x + 20 \)
|
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 54\cdot 109 + 74\cdot 109^{2} + 48\cdot 109^{3} + 50\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 8\cdot 109 + 51\cdot 109^{2} + 45\cdot 109^{3} + 45\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 32 + 15\cdot 109 + 82\cdot 109^{2} + 48\cdot 109^{3} + 58\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 54 + 31\cdot 109 + 10\cdot 109^{2} + 75\cdot 109^{3} + 63\cdot 109^{4} +O(109^{5})\)
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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.