Basic invariants
Dimension: | $3$ |
Group: | $A_4$ |
Conductor: | \(33489\)\(\medspace = 3^{2} \cdot 61^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.4.33489.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4$ |
Parity: | even |
Projective image: | $A_4$ |
Projective field: | Galois closure of 4.4.33489.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 233 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 123 + 43\cdot 233 + 143\cdot 233^{2} + 102\cdot 233^{3} + 125\cdot 233^{4} +O(233^{5})\)
$r_{ 2 }$ |
$=$ |
\( 132 + 64\cdot 233 + 31\cdot 233^{2} + 176\cdot 233^{3} + 204\cdot 233^{4} +O(233^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 216 + 117\cdot 233 + 44\cdot 233^{2} + 179\cdot 233^{3} + 128\cdot 233^{4} +O(233^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 228 + 6\cdot 233 + 14\cdot 233^{2} + 8\cdot 233^{3} + 7\cdot 233^{4} +O(233^{5})\)
| |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$4$ | $3$ | $(1,2,3)$ | $0$ |
$4$ | $3$ | $(1,3,2)$ | $0$ |