Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(459\)\(\medspace = 3^{3} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 6.0.10744731.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Determinant: | 1.51.2t1.a.a |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 6.0.10744731.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 9x^{4} - 3x^{2} + 27x + 33 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a + 24 + \left(2 a + 17\right)\cdot 31 + \left(17 a + 12\right)\cdot 31^{2} + \left(13 a + 16\right)\cdot 31^{3} + \left(21 a + 23\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 21 + \left(9 a + 14\right)\cdot 31 + \left(13 a + 9\right)\cdot 31^{2} + \left(9 a + 13\right)\cdot 31^{3} + \left(2 a + 14\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 6 + \left(28 a + 1\right)\cdot 31 + \left(13 a + 13\right)\cdot 31^{2} + \left(17 a + 26\right)\cdot 31^{3} + \left(9 a + 21\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a + 4 + \left(21 a + 26\right)\cdot 31 + \left(17 a + 26\right)\cdot 31^{2} + \left(21 a + 18\right)\cdot 31^{3} + \left(28 a + 9\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + \left(6 a + 13\right)\cdot 31 + 18\cdot 31^{2} + \left(18 a + 6\right)\cdot 31^{3} + \left(26 a + 25\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 a + 10 + \left(24 a + 20\right)\cdot 31 + \left(30 a + 12\right)\cdot 31^{2} + \left(12 a + 11\right)\cdot 31^{3} + \left(4 a + 29\right)\cdot 31^{4} +O(31^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,2,6)(3,5,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.