Basic invariants
| Dimension: | $3$ |
| Group: | $A_4$ |
| Conductor: | \(223729\)\(\medspace = 11^{2} \cdot 43^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.223729.1 |
| 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 4.4.223729.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 16x^{2} - 7x + 27 \)
|
The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 66\cdot 151 + 144\cdot 151^{2} + 93\cdot 151^{3} + 111\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 + 31\cdot 151 + 88\cdot 151^{2} + 114\cdot 151^{3} + 139\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 127 + 60\cdot 151 + 11\cdot 151^{2} + 84\cdot 151^{3} + 54\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 134 + 143\cdot 151 + 57\cdot 151^{2} + 9\cdot 151^{3} + 147\cdot 151^{4} +O(151^{5})\)
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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 |
| $1$ | $1$ | $()$ | $3$ | ✓ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | |
| $4$ | $3$ | $(1,2,3)$ | $0$ | |
| $4$ | $3$ | $(1,3,2)$ | $0$ |