Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(192589\)\(\medspace = 29^{2} \cdot 229 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.6.292887232721.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | even |
| Determinant: | 1.229.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.3.229.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 176x^{4} - 65x^{3} + 7744x^{2} + 5720x - 80296 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 + 26\cdot 31 + 3\cdot 31^{2} + 10\cdot 31^{3} + 4\cdot 31^{4} + 15\cdot 31^{5} + 26\cdot 31^{6} + 12\cdot 31^{7} + 2\cdot 31^{8} + 15\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a + 8 + \left(28 a + 24\right)\cdot 31 + \left(9 a + 11\right)\cdot 31^{2} + \left(6 a + 15\right)\cdot 31^{3} + \left(22 a + 13\right)\cdot 31^{4} + \left(28 a + 24\right)\cdot 31^{5} + 28\cdot 31^{6} + \left(25 a + 29\right)\cdot 31^{7} + \left(29 a + 21\right)\cdot 31^{8} + \left(28 a + 27\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 a + 3 + \left(10 a + 21\right)\cdot 31 + \left(a + 17\right)\cdot 31^{2} + \left(29 a + 28\right)\cdot 31^{3} + \left(23 a + 3\right)\cdot 31^{4} + 19 a\cdot 31^{5} + \left(19 a + 8\right)\cdot 31^{6} + \left(12 a + 6\right)\cdot 31^{7} + 20\cdot 31^{8} + \left(27 a + 27\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 30 + \left(20 a + 13\right)\cdot 31 + \left(29 a + 9\right)\cdot 31^{2} + \left(a + 23\right)\cdot 31^{3} + \left(7 a + 22\right)\cdot 31^{4} + \left(11 a + 15\right)\cdot 31^{5} + \left(11 a + 27\right)\cdot 31^{6} + \left(18 a + 11\right)\cdot 31^{7} + \left(30 a + 8\right)\cdot 31^{8} + \left(3 a + 19\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 + 4\cdot 31 + 16\cdot 31^{2} + 28\cdot 31^{3} + 27\cdot 31^{4} + 8\cdot 31^{5} + 15\cdot 31^{7} + 14\cdot 31^{8} + 9\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a + 11 + \left(2 a + 2\right)\cdot 31 + \left(21 a + 3\right)\cdot 31^{2} + \left(24 a + 18\right)\cdot 31^{3} + \left(8 a + 20\right)\cdot 31^{4} + \left(2 a + 28\right)\cdot 31^{5} + \left(30 a + 1\right)\cdot 31^{6} + \left(5 a + 17\right)\cdot 31^{7} + \left(a + 25\right)\cdot 31^{8} + \left(2 a + 24\right)\cdot 31^{9} +O(31^{10})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,3)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,4,5,3,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.