Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3696\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.23905728.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.231.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.231.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 6x^{4} + 9x^{2} - 7 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 23\cdot 31 + 17\cdot 31^{2} + 5\cdot 31^{3} + 9\cdot 31^{4} + 8\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 11 + \left(22 a + 18\right)\cdot 31 + \left(9 a + 23\right)\cdot 31^{2} + \left(21 a + 11\right)\cdot 31^{3} + \left(11 a + 25\right)\cdot 31^{4} + \left(25 a + 22\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 4 + \left(8 a + 20\right)\cdot 31 + \left(21 a + 20\right)\cdot 31^{2} + \left(9 a + 13\right)\cdot 31^{3} + \left(19 a + 27\right)\cdot 31^{4} + \left(5 a + 30\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 7\cdot 31 + 13\cdot 31^{2} + 25\cdot 31^{3} + 21\cdot 31^{4} + 22\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 a + 20 + \left(8 a + 12\right)\cdot 31 + \left(21 a + 7\right)\cdot 31^{2} + \left(9 a + 19\right)\cdot 31^{3} + \left(19 a + 5\right)\cdot 31^{4} + \left(5 a + 8\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 27 + \left(22 a + 10\right)\cdot 31 + \left(9 a + 10\right)\cdot 31^{2} + \left(21 a + 17\right)\cdot 31^{3} + \left(11 a + 3\right)\cdot 31^{4} + 25 a\cdot 31^{5} +O(31^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,5,3,4,2,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.