Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(341164\)\(\medspace = 2^{2} \cdot 19 \cdot 67^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.33006934672.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.76.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - 81x^{4} + 133x^{3} + 2894x^{2} - 6922x - 86274 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 + 7\cdot 31 + 9\cdot 31^{2} + 17\cdot 31^{3} + 13\cdot 31^{4} + 27\cdot 31^{5} + 13\cdot 31^{6} + 13\cdot 31^{7} + 15\cdot 31^{8} +O(31^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 + 31 + 6\cdot 31^{2} + 14\cdot 31^{3} + 3\cdot 31^{4} + 4\cdot 31^{5} + 7\cdot 31^{6} + 11\cdot 31^{7} + 25\cdot 31^{8} +O(31^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a + 13 + 18\cdot 31 + \left(14 a + 14\right)\cdot 31^{2} + \left(16 a + 9\right)\cdot 31^{3} + \left(22 a + 13\right)\cdot 31^{4} + \left(a + 11\right)\cdot 31^{5} + \left(8 a + 4\right)\cdot 31^{6} + \left(3 a + 1\right)\cdot 31^{7} + \left(7 a + 16\right)\cdot 31^{8} +O(31^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 22 + \left(26 a + 20\right)\cdot 31 + \left(3 a + 19\right)\cdot 31^{2} + \left(24 a + 20\right)\cdot 31^{3} + \left(17 a + 4\right)\cdot 31^{4} + 6 a\cdot 31^{5} + \left(7 a + 5\right)\cdot 31^{6} + \left(7 a + 30\right)\cdot 31^{7} + \left(29 a + 9\right)\cdot 31^{8} +O(31^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 14 + \left(30 a + 3\right)\cdot 31 + \left(16 a + 11\right)\cdot 31^{2} + \left(14 a + 28\right)\cdot 31^{3} + \left(8 a + 10\right)\cdot 31^{4} + \left(29 a + 23\right)\cdot 31^{5} + \left(22 a + 18\right)\cdot 31^{6} + \left(27 a + 30\right)\cdot 31^{7} + \left(23 a + 26\right)\cdot 31^{8} +O(31^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 a + 24 + \left(4 a + 9\right)\cdot 31 + \left(27 a + 1\right)\cdot 31^{2} + \left(6 a + 3\right)\cdot 31^{3} + \left(13 a + 16\right)\cdot 31^{4} + \left(24 a + 26\right)\cdot 31^{5} + \left(23 a + 12\right)\cdot 31^{6} + \left(23 a + 6\right)\cdot 31^{7} + \left(a + 30\right)\cdot 31^{8} +O(31^{9})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $2$ | $3$ | $(1,4,6)(2,3,5)$ | $-1$ |
| $2$ | $6$ | $(1,5,4,2,6,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.