Basic invariants
Dimension: | $2$ |
Group: | $D_{6}$ |
Conductor: | \(8820\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.518616000.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{6}$ |
Parity: | odd |
Determinant: | 1.20.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.980.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 5x^{4} - 14x^{3} + 55x^{2} - 70x + 34 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( a + 1 + \left(9 a + 4\right)\cdot 11 + \left(9 a + 10\right)\cdot 11^{2} + 5 a\cdot 11^{3} + 7 a\cdot 11^{4} + \left(a + 4\right)\cdot 11^{5} + 5 a\cdot 11^{6} + \left(7 a + 6\right)\cdot 11^{7} + \left(a + 5\right)\cdot 11^{8} +O(11^{9})\) |
$r_{ 2 }$ | $=$ | \( 8 + 4\cdot 11 + 6\cdot 11^{2} + 8\cdot 11^{3} + 7\cdot 11^{4} + 5\cdot 11^{5} + 2\cdot 11^{6} + 3\cdot 11^{7} + 8\cdot 11^{8} +O(11^{9})\) |
$r_{ 3 }$ | $=$ | \( 6 a + 5 + \left(5 a + 4\right)\cdot 11 + \left(8 a + 8\right)\cdot 11^{2} + \left(5 a + 10\right)\cdot 11^{4} + \left(a + 2\right)\cdot 11^{5} + \left(5 a + 5\right)\cdot 11^{6} + \left(8 a + 1\right)\cdot 11^{7} + 10\cdot 11^{8} +O(11^{9})\) |
$r_{ 4 }$ | $=$ | \( 10 a + 5 + \left(a + 6\right)\cdot 11 + \left(a + 7\right)\cdot 11^{2} + \left(5 a + 3\right)\cdot 11^{3} + \left(3 a + 2\right)\cdot 11^{4} + \left(9 a + 3\right)\cdot 11^{5} + \left(5 a + 8\right)\cdot 11^{6} + \left(3 a + 8\right)\cdot 11^{7} + \left(9 a + 4\right)\cdot 11^{8} +O(11^{9})\) |
$r_{ 5 }$ | $=$ | \( 7 + 3\cdot 11 + 7\cdot 11^{2} + 11^{3} + 5\cdot 11^{4} + 2\cdot 11^{5} + 3\cdot 11^{6} + 5\cdot 11^{7} + 10\cdot 11^{8} +O(11^{9})\) |
$r_{ 6 }$ | $=$ | \( 5 a + 7 + \left(5 a + 9\right)\cdot 11 + \left(2 a + 3\right)\cdot 11^{2} + \left(10 a + 6\right)\cdot 11^{3} + \left(5 a + 7\right)\cdot 11^{4} + \left(9 a + 3\right)\cdot 11^{5} + \left(5 a + 2\right)\cdot 11^{6} + \left(2 a + 8\right)\cdot 11^{7} + \left(10 a + 4\right)\cdot 11^{8} +O(11^{9})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-2$ |
$3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$3$ | $2$ | $(1,4)(3,6)$ | $0$ |
$2$ | $3$ | $(1,5,4)(2,3,6)$ | $-1$ |
$2$ | $6$ | $(1,3,5,6,4,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.