Basic invariants
Dimension: | $2$ |
Group: | $D_{6}$ |
Conductor: | \(1800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.1080000.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{6}$ |
Parity: | odd |
Determinant: | 1.8.2t1.b.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.200.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} - x^{4} - 6x^{3} + 6x^{2} + 4x + 4 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \( x^{2} + 6x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 3\cdot 7 + 5\cdot 7^{3} + 5\cdot 7^{4} + 3\cdot 7^{5} +O(7^{7})\) |
$r_{ 2 }$ | $=$ | \( a + 4 + \left(4 a + 2\right)\cdot 7 + 7^{2} + \left(a + 6\right)\cdot 7^{3} + \left(6 a + 1\right)\cdot 7^{4} + \left(2 a + 4\right)\cdot 7^{5} +O(7^{7})\) |
$r_{ 3 }$ | $=$ | \( 6 a + 5 + \left(2 a + 5\right)\cdot 7 + \left(6 a + 4\right)\cdot 7^{2} + \left(5 a + 6\right)\cdot 7^{3} + 6\cdot 7^{4} + 4 a\cdot 7^{5} + \left(6 a + 5\right)\cdot 7^{6} +O(7^{7})\) |
$r_{ 4 }$ | $=$ | \( 4 + 7^{2} + 3\cdot 7^{3} + 6\cdot 7^{4} + 7^{5} + 5\cdot 7^{6} +O(7^{7})\) |
$r_{ 5 }$ | $=$ | \( 4 a + 2 + \left(4 a + 4\right)\cdot 7 + \left(3 a + 3\right)\cdot 7^{2} + \left(a + 4\right)\cdot 7^{3} + \left(3 a + 2\right)\cdot 7^{4} + 5 a\cdot 7^{5} + \left(2 a + 6\right)\cdot 7^{6} +O(7^{7})\) |
$r_{ 6 }$ | $=$ | \( 3 a + 6 + \left(2 a + 4\right)\cdot 7 + \left(3 a + 2\right)\cdot 7^{2} + \left(5 a + 2\right)\cdot 7^{3} + \left(3 a + 4\right)\cdot 7^{4} + \left(a + 2\right)\cdot 7^{5} + \left(4 a + 3\right)\cdot 7^{6} +O(7^{7})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
$3$ | $2$ | $(1,2)(4,5)$ | $0$ |
$3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ |
$2$ | $3$ | $(1,3,2)(4,6,5)$ | $-1$ |
$2$ | $6$ | $(1,6,2,4,3,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.