Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(49744809\)\(\medspace = 3^{2} \cdot 2351^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.4.7053.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | even |
Projective image: | $S_4$ |
Projective field: | Galois closure of 4.4.7053.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 229 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 9 + 190\cdot 229 + 70\cdot 229^{2} + 68\cdot 229^{3} + 161\cdot 229^{4} +O(229^{5})\) |
$r_{ 2 }$ | $=$ | \( 17 + 132\cdot 229 + 214\cdot 229^{2} + 200\cdot 229^{3} + 51\cdot 229^{4} +O(229^{5})\) |
$r_{ 3 }$ | $=$ | \( 77 + 222\cdot 229 + 157\cdot 229^{2} + 103\cdot 229^{3} + 160\cdot 229^{4} +O(229^{5})\) |
$r_{ 4 }$ | $=$ | \( 128 + 142\cdot 229 + 14\cdot 229^{2} + 85\cdot 229^{3} + 84\cdot 229^{4} +O(229^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$6$ | $2$ | $(1,2)$ | $-1$ |
$8$ | $3$ | $(1,2,3)$ | $0$ |
$6$ | $4$ | $(1,2,3,4)$ | $1$ |