Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(136367\)\(\medspace = 7^{2} \cdot 11^{2} \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.5554637011.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.23.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.23.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 37x^{4} + 211x^{3} + 188x^{2} - 3287x + 167315 \)
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The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 35\cdot 43 + 14\cdot 43^{2} + 13\cdot 43^{3} + 7\cdot 43^{4} + 17\cdot 43^{5} + 36\cdot 43^{6} + 41\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 13 + \left(36 a + 36\right)\cdot 43 + \left(41 a + 7\right)\cdot 43^{2} + \left(4 a + 24\right)\cdot 43^{3} + \left(21 a + 29\right)\cdot 43^{4} + \left(35 a + 12\right)\cdot 43^{5} + \left(34 a + 40\right)\cdot 43^{6} + \left(40 a + 32\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 a + 28 + 18\cdot 43 + \left(22 a + 3\right)\cdot 43^{2} + \left(31 a + 10\right)\cdot 43^{3} + \left(12 a + 27\right)\cdot 43^{4} + \left(16 a + 32\right)\cdot 43^{5} + \left(20 a + 22\right)\cdot 43^{6} + \left(13 a + 25\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 38 a + 18 + \left(6 a + 24\right)\cdot 43 + \left(a + 13\right)\cdot 43^{2} + \left(38 a + 30\right)\cdot 43^{3} + \left(21 a + 2\right)\cdot 43^{4} + \left(7 a + 27\right)\cdot 43^{5} + \left(8 a + 39\right)\cdot 43^{6} + \left(2 a + 38\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 15 + \left(42 a + 32\right)\cdot 43 + \left(20 a + 24\right)\cdot 43^{2} + \left(11 a + 19\right)\cdot 43^{3} + \left(30 a + 8\right)\cdot 43^{4} + \left(26 a + 36\right)\cdot 43^{5} + \left(22 a + 26\right)\cdot 43^{6} + \left(29 a + 18\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 + 25\cdot 43 + 21\cdot 43^{2} + 31\cdot 43^{3} + 10\cdot 43^{4} + 3\cdot 43^{5} + 6\cdot 43^{6} + 14\cdot 43^{7} +O(43^{8})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-2$ |
| $3$ | $2$ | $(2,4)(3,5)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,2,5,6,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.