Basic invariants
Dimension: | $2$ |
Group: | $D_{6}$ |
Conductor: | \(57760\)\(\medspace = 2^{5} \cdot 5 \cdot 19^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.6672435200.7 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{6}$ |
Parity: | odd |
Determinant: | 1.40.2t1.b.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.14440.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} - 17x^{4} + 72x^{3} - 8x^{2} - 32x + 544 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 17 + 4\cdot 17^{2} + 6\cdot 17^{4} + 14\cdot 17^{5} + 13\cdot 17^{6} + 8\cdot 17^{7} + 11\cdot 17^{8} +O(17^{9})\) |
$r_{ 2 }$ | $=$ | \( 12 a + 6 + \left(11 a + 15\right)\cdot 17 + \left(8 a + 6\right)\cdot 17^{2} + \left(9 a + 12\right)\cdot 17^{3} + \left(4 a + 10\right)\cdot 17^{4} + \left(6 a + 2\right)\cdot 17^{5} + \left(5 a + 9\right)\cdot 17^{6} + \left(a + 13\right)\cdot 17^{7} + \left(6 a + 12\right)\cdot 17^{8} +O(17^{9})\) |
$r_{ 3 }$ | $=$ | \( 5 a + 1 + \left(5 a + 15\right)\cdot 17 + \left(8 a + 3\right)\cdot 17^{2} + \left(7 a + 13\right)\cdot 17^{3} + \left(12 a + 5\right)\cdot 17^{4} + \left(10 a + 4\right)\cdot 17^{5} + \left(11 a + 8\right)\cdot 17^{6} + \left(15 a + 9\right)\cdot 17^{7} + 10 a\cdot 17^{8} +O(17^{9})\) |
$r_{ 4 }$ | $=$ | \( 12 a + 11 + \left(11 a + 3\right)\cdot 17 + \left(8 a + 11\right)\cdot 17^{2} + \left(9 a + 11\right)\cdot 17^{3} + \left(4 a + 16\right)\cdot 17^{4} + \left(6 a + 10\right)\cdot 17^{5} + \left(5 a + 14\right)\cdot 17^{6} + a\cdot 17^{7} + \left(6 a + 14\right)\cdot 17^{8} +O(17^{9})\) |
$r_{ 5 }$ | $=$ | \( 12 + 12\cdot 17 + 16\cdot 17^{2} + 6\cdot 17^{5} + 8\cdot 17^{6} + 4\cdot 17^{7} + 10\cdot 17^{8} +O(17^{9})\) |
$r_{ 6 }$ | $=$ | \( 5 a + 6 + \left(5 a + 3\right)\cdot 17 + \left(8 a + 8\right)\cdot 17^{2} + \left(7 a + 12\right)\cdot 17^{3} + \left(12 a + 11\right)\cdot 17^{4} + \left(10 a + 12\right)\cdot 17^{5} + \left(11 a + 13\right)\cdot 17^{6} + \left(15 a + 13\right)\cdot 17^{7} + \left(10 a + 1\right)\cdot 17^{8} +O(17^{9})\) |
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 value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-2$ |
$3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
$3$ | $2$ | $(1,6)(3,5)$ | $0$ |
$2$ | $3$ | $(1,4,6)(2,3,5)$ | $-1$ |
$2$ | $6$ | $(1,3,4,5,6,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.