Basic invariants
Dimension: | $9$ |
Group: | $S_6$ |
Conductor: | \(580\!\cdots\!944\)\(\medspace = 2^{18} \cdot 7^{6} \cdot 757^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.339136.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 20T145 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.339136.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} + 2x^{4} - x^{2} - 2x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 a + 10 + \left(5 a + 5\right)\cdot 11 + \left(10 a + 5\right)\cdot 11^{2} + \left(6 a + 8\right)\cdot 11^{3} + \left(9 a + 1\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 2 }$ | $=$ | \( 5 a + 3 + \left(a + 5\right)\cdot 11 + \left(3 a + 9\right)\cdot 11^{2} + \left(10 a + 1\right)\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})\) |
$r_{ 3 }$ | $=$ | \( 6 a + 1 + \left(9 a + 6\right)\cdot 11 + \left(7 a + 9\right)\cdot 11^{2} + 6\cdot 11^{3} + \left(10 a + 2\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 4 }$ | $=$ | \( 6 a + 4 + \left(a + 8\right)\cdot 11 + \left(3 a + 10\right)\cdot 11^{2} + 3 a\cdot 11^{3} + \left(6 a + 4\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 5 }$ | $=$ | \( 3 a + 9 + \left(5 a + 9\right)\cdot 11 + 8\cdot 11^{2} + \left(4 a + 3\right)\cdot 11^{3} + a\cdot 11^{4} +O(11^{5})\) |
$r_{ 6 }$ | $=$ | \( 5 a + 6 + \left(9 a + 8\right)\cdot 11 + \left(7 a + 10\right)\cdot 11^{2} + \left(7 a + 10\right)\cdot 11^{3} + \left(4 a + 3\right)\cdot 11^{4} +O(11^{5})\) |
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 |
$1$ | $1$ | $()$ | $9$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
$15$ | $2$ | $(1,2)$ | $-3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$40$ | $3$ | $(1,2,3)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)$ | $1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.