Basic invariants
Dimension: | $2$ |
Group: | $D_{6}$ |
Conductor: | \(3311\)\(\medspace = 7 \cdot 11 \cdot 43 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.471397003.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{6}$ |
Parity: | odd |
Determinant: | 1.3311.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.3311.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} - 55x^{2} + 64x + 97 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 21 a + 1 + 30 a\cdot 31 + \left(15 a + 22\right)\cdot 31^{2} + \left(20 a + 11\right)\cdot 31^{3} + \left(12 a + 14\right)\cdot 31^{4} + \left(25 a + 25\right)\cdot 31^{5} + \left(4 a + 7\right)\cdot 31^{6} +O(31^{7})\)
$r_{ 2 }$ |
$=$ |
\( 8 + 16\cdot 31 + 2\cdot 31^{2} + 23\cdot 31^{3} + 30\cdot 31^{4} + 20\cdot 31^{5} + 15\cdot 31^{6} +O(31^{7})\)
| $r_{ 3 }$ |
$=$ |
\( 24 + 14\cdot 31 + 28\cdot 31^{2} + 7\cdot 31^{3} + 10\cdot 31^{5} + 15\cdot 31^{6} +O(31^{7})\)
| $r_{ 4 }$ |
$=$ |
\( 10 a + \left(15 a + 9\right)\cdot 31^{2} + \left(10 a + 19\right)\cdot 31^{3} + \left(18 a + 16\right)\cdot 31^{4} + \left(5 a + 5\right)\cdot 31^{5} + \left(26 a + 23\right)\cdot 31^{6} +O(31^{7})\)
| $r_{ 5 }$ |
$=$ |
\( 21 a + 20 + \left(30 a + 21\right)\cdot 31 + \left(15 a + 7\right)\cdot 31^{2} + \left(20 a + 25\right)\cdot 31^{3} + \left(12 a + 11\right)\cdot 31^{4} + \left(25 a + 29\right)\cdot 31^{5} + \left(4 a + 7\right)\cdot 31^{6} +O(31^{7})\)
| $r_{ 6 }$ |
$=$ |
\( 10 a + 12 + 9\cdot 31 + \left(15 a + 23\right)\cdot 31^{2} + \left(10 a + 5\right)\cdot 31^{3} + \left(18 a + 19\right)\cdot 31^{4} + \left(5 a + 1\right)\cdot 31^{5} + \left(26 a + 23\right)\cdot 31^{6} +O(31^{7})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
$3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$3$ | $2$ | $(1,6)(4,5)$ | $0$ |
$2$ | $3$ | $(1,3,6)(2,5,4)$ | $-1$ |
$2$ | $6$ | $(1,5,3,4,6,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.