Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(2563\)\(\medspace = 11 \cdot 233 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.72258659.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.2563.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 40 a + 15 + \left(44 a + 5\right)\cdot 47 + \left(11 a + 31\right)\cdot 47^{2} + \left(7 a + 24\right)\cdot 47^{3} + \left(20 a + 40\right)\cdot 47^{4} + \left(18 a + 17\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 a + \left(44 a + 39\right)\cdot 47 + \left(11 a + 36\right)\cdot 47^{2} + \left(7 a + 19\right)\cdot 47^{3} + \left(20 a + 20\right)\cdot 47^{4} + \left(18 a + 12\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 33 + \left(2 a + 41\right)\cdot 47 + \left(35 a + 15\right)\cdot 47^{2} + \left(39 a + 22\right)\cdot 47^{3} + \left(26 a + 6\right)\cdot 47^{4} + \left(28 a + 29\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 + 34\cdot 47 + 34\cdot 47^{2} + 37\cdot 47^{3} + 42\cdot 47^{4} + 44\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 + 12\cdot 47 + 12\cdot 47^{2} + 9\cdot 47^{3} + 4\cdot 47^{4} + 2\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 1 + \left(2 a + 8\right)\cdot 47 + \left(35 a + 10\right)\cdot 47^{2} + \left(39 a + 27\right)\cdot 47^{3} + \left(26 a + 26\right)\cdot 47^{4} + \left(28 a + 34\right)\cdot 47^{5} +O(47^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(3,4)$ | $0$ |
| $2$ | $3$ | $(1,6,5)(2,4,3)$ | $-1$ |
| $2$ | $6$ | $(1,4,6,3,5,2)$ | $1$ |