Basic invariants
| Dimension: | $3$ |
| Group: | $A_4$ |
| Conductor: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.5184.1 |
| 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 4.0.5184.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 6x^{2} - 4x + 2 \)
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The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 52\cdot 71 + 50\cdot 71^{2} + 36\cdot 71^{3} + 9\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 33 + 42\cdot 71 + 35\cdot 71^{2} + 68\cdot 71^{3} + 3\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 45 + 68\cdot 71 + 5\cdot 71^{2} + 22\cdot 71^{3} + 60\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 63 + 49\cdot 71 + 49\cdot 71^{2} + 14\cdot 71^{3} + 68\cdot 71^{4} +O(71^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
| $4$ | $3$ | $(1,2,3)$ | $0$ | |
| $4$ | $3$ | $(1,3,2)$ | $0$ |