Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(106929\)\(\medspace = 3^{2} \cdot 109^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.2943.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.2943.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 29 + 33\cdot 101 + 100\cdot 101^{2} + 21\cdot 101^{3} + 11\cdot 101^{4} +O(101^{5})\) |
$r_{ 2 }$ | $=$ | \( 44 + 82\cdot 101 + 77\cdot 101^{2} + 44\cdot 101^{3} + 72\cdot 101^{4} +O(101^{5})\) |
$r_{ 3 }$ | $=$ | \( 53 + 15\cdot 101 + 8\cdot 101^{2} + 56\cdot 101^{3} + 71\cdot 101^{4} +O(101^{5})\) |
$r_{ 4 }$ | $=$ | \( 77 + 70\cdot 101 + 15\cdot 101^{2} + 79\cdot 101^{3} + 46\cdot 101^{4} +O(101^{5})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$6$ | $2$ | $(1,2)$ | $-1$ |
$8$ | $3$ | $(1,2,3)$ | $0$ |
$6$ | $4$ | $(1,2,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.