Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(92637\)\(\medspace = 3^{3} \cdot 47 \cdot 73 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.92637.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | even |
Determinant: | 1.10293.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.92637.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} - x^{4} + 2x^{2} - x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 15 a + 12 + \left(8 a + 5\right)\cdot 23 + \left(18 a + 22\right)\cdot 23^{2} + \left(12 a + 4\right)\cdot 23^{3} + \left(18 a + 11\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 2 }$ | $=$ | \( a + 3 + \left(16 a + 20\right)\cdot 23 + \left(19 a + 22\right)\cdot 23^{2} + \left(20 a + 10\right)\cdot 23^{3} + 13 a\cdot 23^{4} +O(23^{5})\) |
$r_{ 3 }$ | $=$ | \( 8 a + 19 + \left(14 a + 7\right)\cdot 23 + \left(4 a + 4\right)\cdot 23^{2} + \left(10 a + 12\right)\cdot 23^{3} + \left(4 a + 12\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 4 }$ | $=$ | \( 19 + 15\cdot 23 + 3\cdot 23^{2} + 11\cdot 23^{3} + 4\cdot 23^{4} +O(23^{5})\) |
$r_{ 5 }$ | $=$ | \( 22 a + 5 + \left(6 a + 5\right)\cdot 23 + 3 a\cdot 23^{2} + \left(2 a + 10\right)\cdot 23^{3} + \left(9 a + 7\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 6 }$ | $=$ | \( 12 + 14\cdot 23 + 15\cdot 23^{2} + 19\cdot 23^{3} + 9\cdot 23^{4} +O(23^{5})\) |
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.