Basic invariants
Dimension: | $2$ |
Group: | $D_{6}$ |
Conductor: | \(2160\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.2.3499200.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{6}$ |
Parity: | odd |
Projective image: | $S_3$ |
Projective field: | Galois closure of 3.1.135.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{2} + 7x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 4 + 8\cdot 11 + 8\cdot 11^{2} + 5\cdot 11^{3} + 7\cdot 11^{4} + 9\cdot 11^{5} + 7\cdot 11^{6} +O(11^{7})\) |
$r_{ 2 }$ | $=$ | \( 8 a + \left(4 a + 3\right)\cdot 11 + 4 a\cdot 11^{2} + \left(9 a + 4\right)\cdot 11^{3} + \left(8 a + 6\right)\cdot 11^{4} + 4 a\cdot 11^{5} + \left(4 a + 9\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 3 }$ | $=$ | \( 3 a + 10 + \left(6 a + 2\right)\cdot 11 + \left(6 a + 2\right)\cdot 11^{2} + \left(a + 4\right)\cdot 11^{3} + \left(2 a + 10\right)\cdot 11^{4} + \left(6 a + 10\right)\cdot 11^{5} + \left(6 a + 10\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 4 }$ | $=$ | \( 7 + 2\cdot 11 + 2\cdot 11^{2} + 5\cdot 11^{3} + 3\cdot 11^{4} + 11^{5} + 3\cdot 11^{6} +O(11^{7})\) |
$r_{ 5 }$ | $=$ | \( 3 a + \left(6 a + 8\right)\cdot 11 + \left(6 a + 10\right)\cdot 11^{2} + \left(a + 6\right)\cdot 11^{3} + \left(2 a + 4\right)\cdot 11^{4} + \left(6 a + 10\right)\cdot 11^{5} + \left(6 a + 1\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 6 }$ | $=$ | \( 8 a + 1 + \left(4 a + 8\right)\cdot 11 + \left(4 a + 8\right)\cdot 11^{2} + \left(9 a + 6\right)\cdot 11^{3} + 8 a\cdot 11^{4} + 4 a\cdot 11^{5} + 4 a\cdot 11^{6} +O(11^{7})\) |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
$3$ | $2$ | $(2,3)(5,6)$ | $0$ |
$3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
$2$ | $3$ | $(1,6,5)(2,4,3)$ | $-1$ |
$2$ | $6$ | $(1,2,6,4,5,3)$ | $1$ |