Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(17064\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 79 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 3.1.17064.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Determinant: | 1.1896.2t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.17064.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} + 21x - 34 \)
|
The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 7 + 3\cdot 7^{2} + 7^{3} +O(7^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 2\cdot 7^{2} + 4\cdot 7^{4} +O(7^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 + 4\cdot 7 + 7^{2} + 5\cdot 7^{3} + 2\cdot 7^{4} +O(7^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,2)$ | $0$ | ✓ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |