Basic invariants
Dimension: | $2$ |
Group: | $D_{6}$ |
Conductor: | \(2855\)\(\medspace = 5 \cdot 571 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.4654235275.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{6}$ |
Parity: | odd |
Determinant: | 1.2855.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.2855.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + 11x^{4} - 23x^{3} + 38x^{2} - 65x + 185 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 17 + 2\cdot 23 + 9\cdot 23^{2} + 5\cdot 23^{3} + 16\cdot 23^{4} + 7\cdot 23^{5} + 9\cdot 23^{6} +O(23^{7})\)
$r_{ 2 }$ |
$=$ |
\( 6 a + 9 + \left(20 a + 4\right)\cdot 23 + \left(6 a + 10\right)\cdot 23^{2} + \left(15 a + 8\right)\cdot 23^{3} + 10 a\cdot 23^{4} + \left(19 a + 5\right)\cdot 23^{5} + \left(9 a + 18\right)\cdot 23^{6} +O(23^{7})\)
| $r_{ 3 }$ |
$=$ |
\( 17 a + 21 + \left(2 a + 15\right)\cdot 23 + \left(16 a + 3\right)\cdot 23^{2} + \left(7 a + 9\right)\cdot 23^{3} + \left(12 a + 6\right)\cdot 23^{4} + \left(3 a + 10\right)\cdot 23^{5} + \left(13 a + 18\right)\cdot 23^{6} +O(23^{7})\)
| $r_{ 4 }$ |
$=$ |
\( 12 a + 16 + 6\cdot 23 + \left(18 a + 22\right)\cdot 23^{2} + \left(3 a + 15\right)\cdot 23^{3} + \left(2 a + 9\right)\cdot 23^{4} + \left(13 a + 7\right)\cdot 23^{5} + \left(5 a + 17\right)\cdot 23^{6} +O(23^{7})\)
| $r_{ 5 }$ |
$=$ |
\( 11 a + 17 + \left(22 a + 18\right)\cdot 23 + \left(4 a + 11\right)\cdot 23^{2} + \left(19 a + 5\right)\cdot 23^{3} + \left(20 a + 10\right)\cdot 23^{4} + \left(9 a + 8\right)\cdot 23^{5} + \left(17 a + 15\right)\cdot 23^{6} +O(23^{7})\)
| $r_{ 6 }$ |
$=$ |
\( 14 + 20\cdot 23 + 11\cdot 23^{2} + 23^{3} + 3\cdot 23^{4} + 7\cdot 23^{5} + 13\cdot 23^{6} +O(23^{7})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-2$ |
$3$ | $2$ | $(1,2)(5,6)$ | $0$ |
$3$ | $2$ | $(1,6)(2,4)(3,5)$ | $0$ |
$2$ | $3$ | $(1,2,3)(4,6,5)$ | $-1$ |
$2$ | $6$ | $(1,4,2,6,3,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.