Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(364\)\(\medspace = 2^{2} \cdot 7 \cdot 13 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 3.1.364.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | odd |
Determinant: | 1.91.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.364.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} + 4x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 7 + 7\cdot 41 + 37\cdot 41^{2} + 2\cdot 41^{3} + 32\cdot 41^{4} +O(41^{5})\) |
$r_{ 2 }$ | $=$ | \( 37 + 29\cdot 41 + 13\cdot 41^{2} + 14\cdot 41^{3} + 24\cdot 41^{4} +O(41^{5})\) |
$r_{ 3 }$ | $=$ | \( 38 + 3\cdot 41 + 31\cdot 41^{2} + 23\cdot 41^{3} + 25\cdot 41^{4} +O(41^{5})\) |
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 value |
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,2)$ | $0$ |
$2$ | $3$ | $(1,2,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.