Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(2149004\)\(\medspace = 2^{2} \cdot 11 \cdot 13^{2} \cdot 17^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.229866063856.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.11.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.44.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 57x^{4} - 553x^{3} + 9462x^{2} - 39766x + 1875886 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 10\cdot 29 + 21\cdot 29^{2} + 8\cdot 29^{3} + 28\cdot 29^{4} + 14\cdot 29^{5} + 28\cdot 29^{6} + 24\cdot 29^{7} + 19\cdot 29^{8} + 22\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a + 25 + \left(10 a + 22\right)\cdot 29 + \left(16 a + 27\right)\cdot 29^{2} + \left(6 a + 16\right)\cdot 29^{3} + \left(25 a + 21\right)\cdot 29^{4} + \left(2 a + 23\right)\cdot 29^{5} + 16\cdot 29^{6} + \left(23 a + 10\right)\cdot 29^{7} + \left(11 a + 18\right)\cdot 29^{8} + \left(23 a + 11\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 14 + \left(18 a + 26\right)\cdot 29 + \left(12 a + 11\right)\cdot 29^{2} + \left(22 a + 4\right)\cdot 29^{3} + \left(3 a + 25\right)\cdot 29^{4} + \left(26 a + 12\right)\cdot 29^{5} + \left(28 a + 14\right)\cdot 29^{6} + \left(5 a + 9\right)\cdot 29^{7} + \left(17 a + 25\right)\cdot 29^{8} + 5 a\cdot 29^{9} +O(29^{10})\)
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| $r_{ 4 }$ | $=$ |
\( 20 + 6\cdot 29 + 17\cdot 29^{2} + 15\cdot 29^{3} + 15\cdot 29^{4} + 5\cdot 29^{5} + 28\cdot 29^{6} + 7\cdot 29^{7} + 22\cdot 29^{8} + 7\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 a + 3 + \left(11 a + 22\right)\cdot 29 + \left(8 a + 3\right)\cdot 29^{2} + \left(a + 7\right)\cdot 29^{3} + \left(27 a + 18\right)\cdot 29^{4} + \left(25 a + 21\right)\cdot 29^{5} + \left(5 a + 26\right)\cdot 29^{6} + \left(4 a + 23\right)\cdot 29^{7} + \left(10 a + 20\right)\cdot 29^{8} + 16 a\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 12 + \left(17 a + 27\right)\cdot 29 + \left(20 a + 4\right)\cdot 29^{2} + \left(27 a + 5\right)\cdot 29^{3} + \left(a + 7\right)\cdot 29^{4} + \left(3 a + 8\right)\cdot 29^{5} + \left(23 a + 1\right)\cdot 29^{6} + \left(24 a + 10\right)\cdot 29^{7} + \left(18 a + 9\right)\cdot 29^{8} + \left(12 a + 14\right)\cdot 29^{9} +O(29^{10})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-2$ |
| $3$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,4,2,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.