Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(940\)\(\medspace = 2^{2} \cdot 5 \cdot 47 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 3.3.940.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | even |
Determinant: | 1.940.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.3.940.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - 7x - 4 \) . |
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 46 + 88\cdot 89 + 48\cdot 89^{2} + 7\cdot 89^{3} + 64\cdot 89^{4} +O(89^{5})\)
$r_{ 2 }$ |
$=$ |
\( 49 + 74\cdot 89 + 7\cdot 89^{2} + 34\cdot 89^{3} + 26\cdot 89^{4} +O(89^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 83 + 14\cdot 89 + 32\cdot 89^{2} + 47\cdot 89^{3} + 87\cdot 89^{4} +O(89^{5})\)
| |
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.