Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(560\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.1568000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.140.1 |
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:
| $r_{ 1 }$ | $=$ |
\( 21 a + 11 + \left(14 a + 20\right)\cdot 23 + \left(7 a + 2\right)\cdot 23^{2} + \left(20 a + 22\right)\cdot 23^{3} + \left(12 a + 1\right)\cdot 23^{4} + \left(9 a + 12\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 + 11\cdot 23 + 5\cdot 23^{2} + 7\cdot 23^{3} + 3\cdot 23^{4} + 10\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a + 16 + \left(14 a + 16\right)\cdot 23 + \left(7 a + 19\right)\cdot 23^{2} + \left(20 a + 13\right)\cdot 23^{3} + \left(12 a + 15\right)\cdot 23^{4} + \left(9 a + 4\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 12 + \left(8 a + 2\right)\cdot 23 + \left(15 a + 20\right)\cdot 23^{2} + 2 a\cdot 23^{3} + \left(10 a + 21\right)\cdot 23^{4} + \left(13 a + 10\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 + 11\cdot 23 + 17\cdot 23^{2} + 15\cdot 23^{3} + 19\cdot 23^{4} + 12\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 7 + \left(8 a + 6\right)\cdot 23 + \left(15 a + 3\right)\cdot 23^{2} + \left(2 a + 9\right)\cdot 23^{3} + \left(10 a + 7\right)\cdot 23^{4} + \left(13 a + 18\right)\cdot 23^{5} +O(23^{6})\)
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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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,2,6)(3,4,5)$ | $-1$ |
| $2$ | $6$ | $(1,3,2,4,6,5)$ | $1$ |