Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(231\)\(\medspace = 3 \cdot 7 \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.586971.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.231.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.231.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 6x^{4} - 4x^{3} + 8x^{2} - 3x + 3 \)
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The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 a + 16 + \left(13 a + 9\right)\cdot 23 + \left(6 a + 12\right)\cdot 23^{2} + \left(15 a + 6\right)\cdot 23^{3} + \left(4 a + 3\right)\cdot 23^{5} +O(23^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 10 + 13\cdot 23 + 21\cdot 23^{2} + 8\cdot 23^{3} + 13\cdot 23^{4} + 9\cdot 23^{5} +O(23^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 8 + 3\cdot 23 + 12\cdot 23^{2} + 20\cdot 23^{4} + 4\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a + 11 + 7 a\cdot 23 + \left(20 a + 12\right)\cdot 23^{2} + \left(7 a + 13\right)\cdot 23^{3} + \left(5 a + 11\right)\cdot 23^{4} + \left(22 a + 12\right)\cdot 23^{5} +O(23^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 3 a + 5 + \left(15 a + 19\right)\cdot 23 + \left(2 a + 21\right)\cdot 23^{2} + \left(15 a + 8\right)\cdot 23^{3} + \left(17 a + 14\right)\cdot 23^{4} + 5\cdot 23^{5} +O(23^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 9 a + 21 + \left(9 a + 22\right)\cdot 23 + \left(16 a + 11\right)\cdot 23^{2} + \left(7 a + 7\right)\cdot 23^{3} + \left(22 a + 9\right)\cdot 23^{4} + \left(18 a + 10\right)\cdot 23^{5} +O(23^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,6,2)(3,4,5)$ | $-1$ |
| $2$ | $6$ | $(1,5,2,4,6,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.