Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(432017\)\(\medspace = 41^{2} \cdot 257 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.6.1169905924153.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | even |
| Determinant: | 1.257.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.3.257.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 267x^{4} + 504x^{3} + 17720x^{2} - 31624x - 249501 \)
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The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 40 a + 8 + \left(17 a + 46\right)\cdot 47 + \left(43 a + 17\right)\cdot 47^{2} + \left(40 a + 30\right)\cdot 47^{3} + \left(44 a + 13\right)\cdot 47^{4} + \left(22 a + 24\right)\cdot 47^{5} + \left(34 a + 16\right)\cdot 47^{6} + \left(22 a + 14\right)\cdot 47^{7} + \left(26 a + 16\right)\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 38 a + 17 + \left(a + 27\right)\cdot 47 + \left(38 a + 30\right)\cdot 47^{2} + \left(8 a + 26\right)\cdot 47^{3} + \left(26 a + 20\right)\cdot 47^{4} + \left(28 a + 3\right)\cdot 47^{5} + 10 a\cdot 47^{6} + \left(37 a + 28\right)\cdot 47^{7} + \left(34 a + 5\right)\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 46 + 5\cdot 47 + 36\cdot 47^{2} + 41\cdot 47^{3} + 17\cdot 47^{4} + 44\cdot 47^{5} + 14\cdot 47^{6} + 7\cdot 47^{7} + 31\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 41 + \left(29 a + 41\right)\cdot 47 + \left(3 a + 39\right)\cdot 47^{2} + \left(6 a + 21\right)\cdot 47^{3} + \left(2 a + 15\right)\cdot 47^{4} + \left(24 a + 25\right)\cdot 47^{5} + \left(12 a + 15\right)\cdot 47^{6} + \left(24 a + 25\right)\cdot 47^{7} + \left(20 a + 46\right)\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 46 + \left(45 a + 39\right)\cdot 47 + \left(8 a + 10\right)\cdot 47^{2} + \left(38 a + 6\right)\cdot 47^{3} + \left(20 a + 17\right)\cdot 47^{4} + \left(18 a + 34\right)\cdot 47^{5} + \left(36 a + 39\right)\cdot 47^{6} + \left(9 a + 44\right)\cdot 47^{7} + \left(12 a + 37\right)\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 32 + 26\cdot 47 + 5\cdot 47^{2} + 14\cdot 47^{3} + 9\cdot 47^{4} + 9\cdot 47^{5} + 7\cdot 47^{6} + 21\cdot 47^{7} + 3\cdot 47^{8} +O(47^{9})\)
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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 value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,3)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,4,5,3,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.