Basic invariants
| Dimension: | $3$ |
| Group: | $A_4$ |
| Conductor: | \(7038409\)\(\medspace = 7^{2} \cdot 379^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.7038409.1 |
| 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 4.4.7038409.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 45x^{2} + 70x - 27 \)
|
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 50\cdot 83 + 72\cdot 83^{2} + 66\cdot 83^{3} + 69\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 + 72\cdot 83 + 59\cdot 83^{2} + 32\cdot 83^{3} + 23\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 59 + 77\cdot 83 + 43\cdot 83^{2} + 72\cdot 83^{3} + 9\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 74 + 48\cdot 83 + 72\cdot 83^{2} + 76\cdot 83^{3} + 62\cdot 83^{4} +O(83^{5})\)
|
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.