Basic invariants
Dimension: | $3$ |
Group: | $A_4$ |
Conductor: | \(246016\)\(\medspace = 2^{8} \cdot 31^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.246016.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4$ |
Parity: | even |
Projective image: | $A_4$ |
Projective field: | Galois closure of 4.0.246016.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 3 + 13\cdot 23 + 13\cdot 23^{2} + 12\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})\)
$r_{ 2 }$ |
$=$ |
\( 9 + 21\cdot 23 + 17\cdot 23^{2} + 4\cdot 23^{3} + 14\cdot 23^{4} +O(23^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 16 + 16\cdot 23 + 19\cdot 23^{2} + 23^{3} + 9\cdot 23^{4} +O(23^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 18 + 17\cdot 23 + 17\cdot 23^{2} + 3\cdot 23^{3} + 3\cdot 23^{4} +O(23^{5})\)
| |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$4$ | $3$ | $(1,2,3)$ | $0$ |
$4$ | $3$ | $(1,3,2)$ | $0$ |