Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(2240\)\(\medspace = 2^{6} \cdot 5 \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.12544000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.35.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.140.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 8x^{4} + 16x^{2} - 10 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 27 + 16\cdot 31 + 29\cdot 31^{2} + 22\cdot 31^{3} + 19\cdot 31^{4} + 15\cdot 31^{5} + 11\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 29 a + \left(20 a + 2\right)\cdot 31 + \left(2 a + 7\right)\cdot 31^{2} + \left(29 a + 30\right)\cdot 31^{3} + \left(12 a + 26\right)\cdot 31^{4} + \left(25 a + 19\right)\cdot 31^{5} + \left(2 a + 15\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 27 + \left(10 a + 14\right)\cdot 31 + \left(28 a + 22\right)\cdot 31^{2} + \left(a + 23\right)\cdot 31^{3} + \left(18 a + 23\right)\cdot 31^{4} + \left(5 a + 26\right)\cdot 31^{5} + \left(28 a + 26\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 + 14\cdot 31 + 31^{2} + 8\cdot 31^{3} + 11\cdot 31^{4} + 15\cdot 31^{5} + 19\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + \left(10 a + 29\right)\cdot 31 + \left(28 a + 23\right)\cdot 31^{2} + a\cdot 31^{3} + \left(18 a + 4\right)\cdot 31^{4} + \left(5 a + 11\right)\cdot 31^{5} + \left(28 a + 15\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 a + 4 + \left(20 a + 16\right)\cdot 31 + \left(2 a + 8\right)\cdot 31^{2} + \left(29 a + 7\right)\cdot 31^{3} + \left(12 a + 7\right)\cdot 31^{4} + \left(25 a + 4\right)\cdot 31^{5} + \left(2 a + 4\right)\cdot 31^{6} +O(31^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,6,5)(2,4,3)$ | $-1$ |
| $2$ | $6$ | $(1,2,6,4,5,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.