Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(104\)\(\medspace = 2^{3} \cdot 13 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 3.1.104.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | odd |
Determinant: | 1.104.2t1.b.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.104.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 14 + 13\cdot 31 + 8\cdot 31^{3} + 19\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$ |
$=$ |
\( 21 + 13\cdot 31 + 30\cdot 31^{2} + 14\cdot 31^{3} + 22\cdot 31^{4} +O(31^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 27 + 3\cdot 31 + 8\cdot 31^{3} + 20\cdot 31^{4} +O(31^{5})\)
| |
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.