Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(756\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 3.1.756.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Determinant: | 1.84.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.756.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - 6x - 12 \)
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The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 11\cdot 17 + 7\cdot 17^{2} + 15\cdot 17^{3} + 14\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 10\cdot 17^{2} + 14\cdot 17^{3} + 8\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 5\cdot 17 + 16\cdot 17^{2} + 3\cdot 17^{3} + 10\cdot 17^{4} +O(17^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,2)$ | $0$ | ✓ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |