Basic invariants
Dimension: | $2$ |
Group: | $D_{6}$ |
Conductor: | \(325703\)\(\medspace = 7^{2} \cdot 17^{2} \cdot 23 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.20503329553.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{6}$ |
Parity: | odd |
Determinant: | 1.23.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.23.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + 61x^{4} - 328x^{3} + 1168x^{2} - 8040x - 597869 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 10 a + 3 + \left(3 a + 9\right)\cdot 11 + \left(3 a + 9\right)\cdot 11^{2} + \left(6 a + 7\right)\cdot 11^{3} + \left(a + 7\right)\cdot 11^{4} + \left(4 a + 10\right)\cdot 11^{5} + \left(7 a + 6\right)\cdot 11^{6} + \left(3 a + 8\right)\cdot 11^{7} + \left(8 a + 10\right)\cdot 11^{8} + \left(7 a + 2\right)\cdot 11^{9} +O(11^{10})\)
$r_{ 2 }$ |
$=$ |
\( 8 a + 10 + 5\cdot 11 + \left(6 a + 8\right)\cdot 11^{2} + 9 a\cdot 11^{3} + \left(2 a + 2\right)\cdot 11^{4} + 7 a\cdot 11^{5} + 2\cdot 11^{6} + \left(3 a + 9\right)\cdot 11^{7} + \left(5 a + 5\right)\cdot 11^{8} + \left(10 a + 1\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 4 + 4\cdot 11 + 3\cdot 11^{2} + 10\cdot 11^{3} + 4\cdot 11^{4} + 6\cdot 11^{5} + 3\cdot 11^{7} + 3\cdot 11^{8} + 4\cdot 11^{9} +O(11^{10})\)
| $r_{ 4 }$ |
$=$ |
\( a + 10 + \left(7 a + 3\right)\cdot 11 + \left(7 a + 8\right)\cdot 11^{2} + \left(4 a + 7\right)\cdot 11^{3} + \left(9 a + 7\right)\cdot 11^{4} + \left(6 a + 3\right)\cdot 11^{5} + \left(3 a + 10\right)\cdot 11^{6} + \left(7 a + 4\right)\cdot 11^{7} + \left(2 a + 7\right)\cdot 11^{8} + \left(3 a + 3\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 3 a + 9 + 10 a\cdot 11 + \left(4 a + 10\right)\cdot 11^{2} + \left(a + 10\right)\cdot 11^{3} + \left(8 a + 3\right)\cdot 11^{4} + \left(3 a + 4\right)\cdot 11^{5} + \left(10 a + 8\right)\cdot 11^{6} + \left(7 a + 9\right)\cdot 11^{7} + \left(5 a + 1\right)\cdot 11^{8} + 5\cdot 11^{9} +O(11^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 10 + 8\cdot 11 + 3\cdot 11^{2} + 6\cdot 11^{3} + 6\cdot 11^{4} + 7\cdot 11^{5} + 4\cdot 11^{6} + 8\cdot 11^{7} + 3\cdot 11^{8} + 4\cdot 11^{9} +O(11^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-2$ |
$3$ | $2$ | $(3,5)(4,6)$ | $0$ |
$3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$2$ | $3$ | $(1,4,6)(2,5,3)$ | $-1$ |
$2$ | $6$ | $(1,3,4,2,6,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.