Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 3.1.648.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | odd |
Determinant: | 1.8.2t1.b.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.648.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - 3x - 10 \) . |
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 15 + 78\cdot 83 + 19\cdot 83^{2} + 19\cdot 83^{3} + 78\cdot 83^{4} +O(83^{5})\)
$r_{ 2 }$ |
$=$ |
\( 75 + 57\cdot 83 + 51\cdot 83^{2} + 26\cdot 83^{3} + 75\cdot 83^{4} +O(83^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 76 + 29\cdot 83 + 11\cdot 83^{2} + 37\cdot 83^{3} + 12\cdot 83^{4} +O(83^{5})\)
| |
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.