Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(1765\)\(\medspace = 5 \cdot 353 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.1765.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | even |
Determinant: | 1.1765.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.0.1765.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - x^{2} - x + 3 \) . |
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 14 + 18\cdot 173 + 26\cdot 173^{2} + 168\cdot 173^{3} + 58\cdot 173^{4} +O(173^{5})\) |
$r_{ 2 }$ | $=$ | \( 52 + 100\cdot 173 + 46\cdot 173^{2} + 123\cdot 173^{3} + 162\cdot 173^{4} +O(173^{5})\) |
$r_{ 3 }$ | $=$ | \( 131 + 120\cdot 173 + 45\cdot 173^{2} + 56\cdot 173^{3} + 155\cdot 173^{4} +O(173^{5})\) |
$r_{ 4 }$ | $=$ | \( 150 + 106\cdot 173 + 54\cdot 173^{2} + 171\cdot 173^{3} + 141\cdot 173^{4} +O(173^{5})\) |
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 | Complex conjugation |
$1$ | $1$ | $()$ | $3$ | |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
$6$ | $2$ | $(1,2)$ | $1$ | |
$8$ | $3$ | $(1,2,3)$ | $0$ | |
$6$ | $4$ | $(1,2,3,4)$ | $-1$ |