Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.7200000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.300.1 |
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:
| $r_{ 1 }$ | $=$ |
\( 19 a + 6 + \left(6 a + 18\right)\cdot 23 + \left(21 a + 6\right)\cdot 23^{2} + \left(8 a + 7\right)\cdot 23^{3} + \left(20 a + 15\right)\cdot 23^{4} + \left(2 a + 10\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 + 14\cdot 23 + 19\cdot 23^{2} + 11\cdot 23^{3} + 6\cdot 23^{4} + 16\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a + 8 + \left(11 a + 17\right)\cdot 23 + \left(13 a + 8\right)\cdot 23^{2} + \left(9 a + 3\right)\cdot 23^{3} + \left(9 a + 6\right)\cdot 23^{4} + \left(8 a + 16\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 21 + \left(16 a + 12\right)\cdot 23 + \left(a + 19\right)\cdot 23^{2} + \left(14 a + 3\right)\cdot 23^{3} + \left(2 a + 1\right)\cdot 23^{4} + \left(20 a + 19\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 2 + \left(11 a + 21\right)\cdot 23 + 9 a\cdot 23^{2} + \left(13 a + 9\right)\cdot 23^{3} + \left(13 a + 15\right)\cdot 23^{4} + 14 a\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 + 7\cdot 23 + 13\cdot 23^{2} + 10\cdot 23^{3} + 23^{4} + 6\cdot 23^{5} +O(23^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,4,2)(3,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,5,2,3,4,6)$ | $1$ |