Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(37423\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.37423.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | odd |
Determinant: | 1.37423.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.0.37423.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + x^{4} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 22 a + 23 + \left(17 a + 39\right)\cdot 41 + \left(5 a + 6\right)\cdot 41^{2} + \left(36 a + 10\right)\cdot 41^{3} + \left(17 a + 37\right)\cdot 41^{4} +O(41^{5})\)
$r_{ 2 }$ |
$=$ |
\( 6 a + 6 + \left(13 a + 10\right)\cdot 41 + \left(12 a + 4\right)\cdot 41^{2} + \left(37 a + 21\right)\cdot 41^{3} + \left(5 a + 9\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 19 a + 7 + \left(23 a + 29\right)\cdot 41 + \left(35 a + 5\right)\cdot 41^{2} + \left(4 a + 31\right)\cdot 41^{3} + \left(23 a + 13\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 34 + 20\cdot 41 + 2\cdot 41^{2} + 30\cdot 41^{3} + 10\cdot 41^{4} +O(41^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 30 + 20\cdot 41 + 34\cdot 41^{2} + 32\cdot 41^{3} + 20\cdot 41^{4} +O(41^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 35 a + 24 + \left(27 a + 2\right)\cdot 41 + \left(28 a + 28\right)\cdot 41^{2} + \left(3 a + 38\right)\cdot 41^{3} + \left(35 a + 30\right)\cdot 41^{4} +O(41^{5})\)
| |
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.