Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(152\)\(\medspace = 2^{3} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.184832.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.152.2t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.152.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 2x^{4} - 2x^{3} - 2x^{2} - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 a + 19 + \left(29 a + 4\right)\cdot 31 + \left(30 a + 25\right)\cdot 31^{2} + \left(24 a + 4\right)\cdot 31^{3} + \left(30 a + 27\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 19 + \left(3 a + 28\right)\cdot 31 + \left(4 a + 26\right)\cdot 31^{2} + \left(25 a + 2\right)\cdot 31^{3} + \left(18 a + 1\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 24 + \left(a + 14\right)\cdot 31 + 26\cdot 31^{2} + \left(6 a + 23\right)\cdot 31^{3} + 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 + 3\cdot 31 + 6\cdot 31^{2} + 23\cdot 31^{3} + 8\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 + 15\cdot 31 + 7\cdot 31^{2} + 20\cdot 31^{3} + 9\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a + 6 + \left(27 a + 26\right)\cdot 31 + 26 a\cdot 31^{2} + \left(5 a + 18\right)\cdot 31^{3} + \left(12 a + 13\right)\cdot 31^{4} +O(31^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,6)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,4,6)$ | $-1$ |
| $2$ | $6$ | $(1,4,3,2,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.