Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(621\)\(\medspace = 3^{3} \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 3.3.621.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | even |
| Determinant: | 1.69.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.3.621.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - 6x - 3 \)
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The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 + 43\cdot 83 + 68\cdot 83^{2} + 70\cdot 83^{3} + 7\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 65 + 5\cdot 83 + 62\cdot 83^{2} + 11\cdot 83^{3} + 2\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 66 + 33\cdot 83 + 35\cdot 83^{2} + 73\cdot 83^{4} +O(83^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $3$ | $2$ | $(1,2)$ | $0$ | |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |