Basic invariants
Dimension: | $3$ |
Group: | $S_4\times C_2$ |
Conductor: | \(28037\)\(\medspace = 23^{2} \cdot 53 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.28037.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4\times C_2$ |
Parity: | even |
Determinant: | 1.53.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.64607.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + 3x^{3} - 2x - 1 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 14 a + 5 + \left(15 a + 9\right)\cdot 17 + \left(14 a + 15\right)\cdot 17^{2} + \left(a + 8\right)\cdot 17^{3} + \left(5 a + 5\right)\cdot 17^{4} + \left(11 a + 13\right)\cdot 17^{5} + \left(5 a + 8\right)\cdot 17^{6} + \left(a + 7\right)\cdot 17^{7} + \left(2 a + 9\right)\cdot 17^{8} +O(17^{9})\)
$r_{ 2 }$ |
$=$ |
\( 16 a + 5 + \left(7 a + 11\right)\cdot 17 + \left(2 a + 7\right)\cdot 17^{2} + \left(9 a + 7\right)\cdot 17^{3} + \left(11 a + 16\right)\cdot 17^{4} + \left(9 a + 11\right)\cdot 17^{5} + \left(8 a + 13\right)\cdot 17^{6} + \left(15 a + 15\right)\cdot 17^{7} + \left(2 a + 5\right)\cdot 17^{8} +O(17^{9})\)
| $r_{ 3 }$ |
$=$ |
\( 14 + 12\cdot 17 + 8\cdot 17^{2} + 9\cdot 17^{4} + 13\cdot 17^{5} + 10\cdot 17^{6} + 16\cdot 17^{7} + 2\cdot 17^{8} +O(17^{9})\)
| $r_{ 4 }$ |
$=$ |
\( a + 4 + \left(9 a + 3\right)\cdot 17 + \left(14 a + 2\right)\cdot 17^{2} + \left(7 a + 14\right)\cdot 17^{3} + \left(5 a + 1\right)\cdot 17^{4} + \left(7 a + 10\right)\cdot 17^{5} + \left(8 a + 12\right)\cdot 17^{6} + \left(a + 5\right)\cdot 17^{7} + \left(14 a + 10\right)\cdot 17^{8} +O(17^{9})\)
| $r_{ 5 }$ |
$=$ |
\( 3 a + 2 + \left(a + 11\right)\cdot 17 + \left(2 a + 14\right)\cdot 17^{2} + \left(15 a + 12\right)\cdot 17^{3} + \left(11 a + 8\right)\cdot 17^{4} + \left(5 a + 2\right)\cdot 17^{5} + \left(11 a + 3\right)\cdot 17^{6} + \left(15 a + 3\right)\cdot 17^{7} + \left(14 a + 10\right)\cdot 17^{8} +O(17^{9})\)
| $r_{ 6 }$ |
$=$ |
\( 6 + 3\cdot 17 + 2\cdot 17^{2} + 7\cdot 17^{3} + 9\cdot 17^{4} + 16\cdot 17^{5} + 17^{6} + 2\cdot 17^{7} + 12\cdot 17^{8} +O(17^{9})\)
| |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
$3$ | $2$ | $(1,2)(3,6)$ | $-1$ |
$3$ | $2$ | $(3,6)$ | $1$ |
$6$ | $2$ | $(1,3)(2,6)$ | $-1$ |
$6$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
$8$ | $3$ | $(1,4,3)(2,5,6)$ | $0$ |
$6$ | $4$ | $(1,6,2,3)$ | $-1$ |
$6$ | $4$ | $(1,2)(3,5,6,4)$ | $1$ |
$8$ | $6$ | $(1,4,3,2,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.