Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(229\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 3.3.229.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | even |
| Determinant: | 1.229.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.3.229.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - 4x - 1 \)
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The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 + 2\cdot 37 + 14\cdot 37^{2} + 35\cdot 37^{3} + 18\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 5\cdot 37 + 3\cdot 37^{2} + 5\cdot 37^{3} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 + 28\cdot 37 + 19\cdot 37^{2} + 33\cdot 37^{3} + 17\cdot 37^{4} +O(37^{5})\)
|
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.