Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(32171\)\(\medspace = 53 \cdot 607 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.32171.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | odd |
Determinant: | 1.32171.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.0.32171.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + 2x^{4} - x^{3} + x^{2} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 20 a + 33 + \left(8 a + 9\right)\cdot 47 + \left(21 a + 38\right)\cdot 47^{2} + 9\cdot 47^{3} + \left(24 a + 21\right)\cdot 47^{4} +O(47^{5})\)
$r_{ 2 }$ |
$=$ |
\( 17 a + 41 + \left(42 a + 29\right)\cdot 47 + \left(5 a + 43\right)\cdot 47^{2} + \left(7 a + 42\right)\cdot 47^{3} + \left(a + 7\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 27 a + 26 + \left(38 a + 6\right)\cdot 47 + \left(25 a + 25\right)\cdot 47^{2} + \left(46 a + 36\right)\cdot 47^{3} + \left(22 a + 21\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 16 + 3\cdot 47 + 39\cdot 47^{2} + 17\cdot 47^{3} + 31\cdot 47^{4} +O(47^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 30 a + 28 + \left(4 a + 3\right)\cdot 47 + \left(41 a + 13\right)\cdot 47^{2} + \left(39 a + 4\right)\cdot 47^{3} + \left(45 a + 3\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 45 + 40\cdot 47 + 28\cdot 47^{2} + 29\cdot 47^{3} + 8\cdot 47^{4} +O(47^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $5$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$40$ | $3$ | $(1,2,3)$ | $2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)$ | $1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.