Basic invariants
Dimension: | $3$ |
Group: | $A_4$ |
Conductor: | \(962361\)\(\medspace = 3^{4} \cdot 109^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.962361.1 |
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 4.0.962361.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 21x^{2} - 24x + 249 \) . |
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 11 + 21\cdot 107 + 99\cdot 107^{2} + 57\cdot 107^{3} + 20\cdot 107^{4} +O(107^{5})\) |
$r_{ 2 }$ | $=$ | \( 46 + 54\cdot 107 + 72\cdot 107^{2} + 97\cdot 107^{3} + 53\cdot 107^{4} +O(107^{5})\) |
$r_{ 3 }$ | $=$ | \( 61 + 78\cdot 107 + 56\cdot 107^{2} + 82\cdot 107^{3} + 35\cdot 107^{4} +O(107^{5})\) |
$r_{ 4 }$ | $=$ | \( 97 + 59\cdot 107 + 92\cdot 107^{2} + 82\cdot 107^{3} + 103\cdot 107^{4} +O(107^{5})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$4$ | $3$ | $(1,2,3)$ | $0$ |
$4$ | $3$ | $(1,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.