Basic invariants
Dimension: | $3$ |
Group: | $A_4$ |
Conductor: | \(480249\)\(\medspace = 3^{4} \cdot 7^{2} \cdot 11^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.480249.1 |
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 4.0.480249.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 3x^{2} - 30x + 75 \) . |
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 24 + 81\cdot 131 + 5\cdot 131^{2} + 55\cdot 131^{3} + 73\cdot 131^{4} +O(131^{5})\) |
$r_{ 2 }$ | $=$ | \( 38 + 61\cdot 131 + 30\cdot 131^{2} + 94\cdot 131^{3} + 2\cdot 131^{4} +O(131^{5})\) |
$r_{ 3 }$ | $=$ | \( 77 + 75\cdot 131 + 76\cdot 131^{2} + 75\cdot 131^{3} + 91\cdot 131^{4} +O(131^{5})\) |
$r_{ 4 }$ | $=$ | \( 124 + 43\cdot 131 + 18\cdot 131^{2} + 37\cdot 131^{3} + 94\cdot 131^{4} +O(131^{5})\) |
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 value | Complex conjugation |
$1$ | $1$ | $()$ | $3$ | |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
$4$ | $3$ | $(1,2,3)$ | $0$ | |
$4$ | $3$ | $(1,3,2)$ | $0$ |