Basic invariants
Dimension: | $2$ |
Group: | $D_{6}$ |
Conductor: | \(891\)\(\medspace = 3^{4} \cdot 11 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.0.2381643.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{6}$ |
Parity: | odd |
Projective image: | $S_3$ |
Projective field: | Galois closure of 3.1.891.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$:
\( x^{2} + 6x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 2\cdot 7 + 4\cdot 7^{2} + 3\cdot 7^{3} + 7^{5} + 3\cdot 7^{6} + 2\cdot 7^{7} + 2\cdot 7^{8} +O(7^{9})\)
$r_{ 2 }$ |
$=$ |
\( 3 a + 1 + \left(a + 3\right)\cdot 7 + \left(a + 1\right)\cdot 7^{2} + \left(2 a + 1\right)\cdot 7^{3} + a\cdot 7^{4} + \left(6 a + 4\right)\cdot 7^{5} + \left(4 a + 2\right)\cdot 7^{6} + \left(4 a + 2\right)\cdot 7^{7} + \left(4 a + 2\right)\cdot 7^{8} +O(7^{9})\)
| $r_{ 3 }$ |
$=$ |
\( 6 + 6\cdot 7^{2} + 5\cdot 7^{3} + 6\cdot 7^{4} + 7^{5} + 2\cdot 7^{6} + 5\cdot 7^{7} +O(7^{9})\)
| $r_{ 4 }$ |
$=$ |
\( 2 a + 3 + \left(2 a + 6\right)\cdot 7 + \left(6 a + 1\right)\cdot 7^{2} + 3\cdot 7^{3} + \left(3 a + 2\right)\cdot 7^{4} + \left(4 a + 5\right)\cdot 7^{5} + \left(4 a + 5\right)\cdot 7^{6} + 2\cdot 7^{7} + \left(2 a + 2\right)\cdot 7^{8} +O(7^{9})\)
| $r_{ 5 }$ |
$=$ |
\( 4 a + 4 + \left(5 a + 1\right)\cdot 7 + \left(5 a + 1\right)\cdot 7^{2} + \left(4 a + 2\right)\cdot 7^{3} + \left(5 a + 6\right)\cdot 7^{4} + 7^{5} + \left(2 a + 1\right)\cdot 7^{6} + \left(2 a + 2\right)\cdot 7^{7} + \left(2 a + 2\right)\cdot 7^{8} +O(7^{9})\)
| $r_{ 6 }$ |
$=$ |
\( 5 a + 5 + \left(4 a + 6\right)\cdot 7 + 5\cdot 7^{2} + \left(6 a + 4\right)\cdot 7^{3} + \left(3 a + 4\right)\cdot 7^{4} + \left(2 a + 6\right)\cdot 7^{5} + \left(2 a + 5\right)\cdot 7^{6} + \left(6 a + 5\right)\cdot 7^{7} + \left(4 a + 3\right)\cdot 7^{8} +O(7^{9})\)
| |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-2$ |
$3$ | $2$ | $(1,2)(3,4)$ | $0$ |
$3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
$2$ | $3$ | $(1,5,2)(3,6,4)$ | $-1$ |
$2$ | $6$ | $(1,6,2,3,5,4)$ | $1$ |