Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.1548800.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.440.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( a + \left(5 a + 10\right)\cdot 23 + \left(10 a + 3\right)\cdot 23^{2} + \left(16 a + 1\right)\cdot 23^{3} + \left(18 a + 2\right)\cdot 23^{4} + 16\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 a + 2 + \left(17 a + 19\right)\cdot 23 + \left(12 a + 18\right)\cdot 23^{2} + 6 a\cdot 23^{3} + 4 a\cdot 23^{4} + \left(22 a + 22\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 + 6\cdot 23 + 22\cdot 23^{2} + 23^{3} + 2\cdot 23^{4} + 15\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 a + \left(17 a + 13\right)\cdot 23 + \left(12 a + 19\right)\cdot 23^{2} + \left(6 a + 21\right)\cdot 23^{3} + \left(4 a + 20\right)\cdot 23^{4} + \left(22 a + 6\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 21 + \left(5 a + 3\right)\cdot 23 + \left(10 a + 4\right)\cdot 23^{2} + \left(16 a + 22\right)\cdot 23^{3} + \left(18 a + 22\right)\cdot 23^{4} +O(23^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 + 16\cdot 23 + 21\cdot 23^{3} + 20\cdot 23^{4} + 7\cdot 23^{5} +O(23^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,6,2)(3,5,4)$ | $-1$ |
| $2$ | $6$ | $(1,5,6,4,2,3)$ | $1$ |