Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(1016\)\(\medspace = 2^{3} \cdot 127 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.1016.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | even |
Projective image: | $S_4$ |
Projective field: | Galois closure of 4.0.1016.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 48 + 75\cdot 173 + 91\cdot 173^{2} + 119\cdot 173^{3} + 156\cdot 173^{4} +O(173^{5})\) |
$r_{ 2 }$ | $=$ | \( 57 + 15\cdot 173 + 94\cdot 173^{2} + 64\cdot 173^{3} + 143\cdot 173^{4} +O(173^{5})\) |
$r_{ 3 }$ | $=$ | \( 73 + 28\cdot 173 + 88\cdot 173^{2} + 171\cdot 173^{3} + 2\cdot 173^{4} +O(173^{5})\) |
$r_{ 4 }$ | $=$ | \( 169 + 53\cdot 173 + 72\cdot 173^{2} + 163\cdot 173^{3} + 42\cdot 173^{4} +O(173^{5})\) |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$6$ | $2$ | $(1,2)$ | $1$ |
$8$ | $3$ | $(1,2,3)$ | $0$ |
$6$ | $4$ | $(1,2,3,4)$ | $-1$ |