Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(148\)\(\medspace = 2^{2} \cdot 37 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 3.3.148.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | even |
Determinant: | 1.37.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.3.148.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - x^{2} - 3x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 10 + 6\cdot 67 + 45\cdot 67^{2} + 30\cdot 67^{3} + 50\cdot 67^{4} +O(67^{5})\)
$r_{ 2 }$ |
$=$ |
\( 62 + 17\cdot 67 + 27\cdot 67^{2} + 39\cdot 67^{3} + 14\cdot 67^{4} +O(67^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 63 + 42\cdot 67 + 61\cdot 67^{2} + 63\cdot 67^{3} + 67^{4} +O(67^{5})\)
| |
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.