Basic invariants
Dimension: | $3$ |
Group: | $A_4$ |
Conductor: | \(165649\)\(\medspace = 11^{2} \cdot 37^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.4.165649.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4$ |
Parity: | even |
Projective image: | $A_4$ |
Projective field: | Galois closure of 4.4.165649.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 8 + 9\cdot 47 + 7\cdot 47^{2} + 16\cdot 47^{3} + 41\cdot 47^{4} +O(47^{5})\) |
$r_{ 2 }$ | $=$ | \( 18 + 17\cdot 47 + 15\cdot 47^{2} + 6\cdot 47^{3} + 42\cdot 47^{4} +O(47^{5})\) |
$r_{ 3 }$ | $=$ | \( 28 + 9\cdot 47 + 31\cdot 47^{2} + 36\cdot 47^{3} + 38\cdot 47^{4} +O(47^{5})\) |
$r_{ 4 }$ | $=$ | \( 42 + 10\cdot 47 + 40\cdot 47^{2} + 34\cdot 47^{3} + 18\cdot 47^{4} +O(47^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$4$ | $3$ | $(1,2,3)$ | $0$ |
$4$ | $3$ | $(1,3,2)$ | $0$ |