Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(1207801\)\(\medspace = 7^{2} \cdot 157^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.2.1099.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | even |
Projective image: | $S_4$ |
Projective field: | Galois closure of 4.2.1099.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 317 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 116 + 117\cdot 317 + 120\cdot 317^{2} + 83\cdot 317^{3} + 188\cdot 317^{4} +O(317^{5})\) |
$r_{ 2 }$ | $=$ | \( 168 + 199\cdot 317 + 307\cdot 317^{2} + 80\cdot 317^{3} + 173\cdot 317^{4} +O(317^{5})\) |
$r_{ 3 }$ | $=$ | \( 170 + 64\cdot 317 + 126\cdot 317^{2} + 41\cdot 317^{3} + 56\cdot 317^{4} +O(317^{5})\) |
$r_{ 4 }$ | $=$ | \( 181 + 252\cdot 317 + 79\cdot 317^{2} + 111\cdot 317^{3} + 216\cdot 317^{4} +O(317^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$6$ | $2$ | $(1,2)$ | $-1$ |
$8$ | $3$ | $(1,2,3)$ | $0$ |
$6$ | $4$ | $(1,2,3,4)$ | $1$ |