Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(1373\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 3.3.1373.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | even |
Determinant: | 1.1373.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.3.1373.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - 8x - 5 \) . |
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 14 + 9\cdot 37 + 21\cdot 37^{2} + 7\cdot 37^{3} + 6\cdot 37^{4} +O(37^{5})\)
$r_{ 2 }$ |
$=$ |
\( 27 + 2\cdot 37 + 3\cdot 37^{2} + 24\cdot 37^{3} + 36\cdot 37^{4} +O(37^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 33 + 24\cdot 37 + 12\cdot 37^{2} + 5\cdot 37^{3} + 31\cdot 37^{4} +O(37^{5})\)
| |
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.