Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(1485961\)\(\medspace = 23^{2} \cdot 53^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.64607.1 |
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 4.2.64607.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - x^{2} - 6x - 17 \) . |
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 10 + 138\cdot 173 + 76\cdot 173^{2} + 150\cdot 173^{3} + 142\cdot 173^{4} +O(173^{5})\)
$r_{ 2 }$ |
$=$ |
\( 43 + 143\cdot 173 + 19\cdot 173^{2} + 148\cdot 173^{3} + 126\cdot 173^{4} +O(173^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 138 + 145\cdot 173 + 120\cdot 173^{2} + 173^{3} + 146\cdot 173^{4} +O(173^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 156 + 91\cdot 173 + 128\cdot 173^{2} + 45\cdot 173^{3} + 103\cdot 173^{4} +O(173^{5})\)
| |
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.