Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(175\)\(\medspace = 5^{2} \cdot 7 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 3.1.175.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | odd |
Projective image: | $S_3$ |
Projective field: | Galois closure of 3.1.175.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 4 + 29\cdot 53 + 24\cdot 53^{2} + 18\cdot 53^{3} + 25\cdot 53^{4} +O(53^{5})\)
$r_{ 2 }$ |
$=$ |
\( 16 + 5\cdot 53 + 20\cdot 53^{2} + 32\cdot 53^{3} + 49\cdot 53^{4} +O(53^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 34 + 18\cdot 53 + 8\cdot 53^{2} + 2\cdot 53^{3} + 31\cdot 53^{4} +O(53^{5})\)
| |
Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ | Character values |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,2)$ | $0$ |
$2$ | $3$ | $(1,2,3)$ | $-1$ |