Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(231\)\(\medspace = 3 \cdot 7 \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.373527.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.231.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 20 + 17\cdot 23 + 9\cdot 23^{2} + 13\cdot 23^{3} + 10\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 14 + \left(22 a + 5\right)\cdot 23 + \left(20 a + 2\right)\cdot 23^{2} + \left(3 a + 1\right)\cdot 23^{3} + 11 a\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a + 3 + 21\cdot 23 + \left(2 a + 21\right)\cdot 23^{2} + \left(19 a + 10\right)\cdot 23^{3} + \left(11 a + 18\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 4 + \left(18 a + 5\right)\cdot 23 + 13\cdot 23^{2} + 16\cdot 23^{3} + 22\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 a + 14 + \left(4 a + 13\right)\cdot 23 + \left(22 a + 19\right)\cdot 23^{2} + \left(22 a + 15\right)\cdot 23^{3} + \left(22 a + 22\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 + 5\cdot 23 + 2\cdot 23^{2} + 11\cdot 23^{3} + 17\cdot 23^{4} +O(23^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-2$ |
| $3$ | $2$ | $(1,2)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,5,3,6,2,4)$ | $1$ |