Basic invariants
Dimension: | $9$ |
Group: | $A_6$ |
Conductor: | \(6499837226778624\)\(\medspace = 2^{24} \cdot 3^{18} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.107495424.3 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PSL(2,9)$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $A_6$ |
Projective stem field: | Galois closure of 6.2.107495424.3 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 12x^{3} + 21x^{2} + 12x - 34 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 4 a + 30 + \left(18 a + 31\right)\cdot 47 + \left(2 a + 30\right)\cdot 47^{2} + \left(14 a + 4\right)\cdot 47^{3} + \left(6 a + 22\right)\cdot 47^{4} + \left(34 a + 16\right)\cdot 47^{5} +O(47^{6})\)
$r_{ 2 }$ |
$=$ |
\( 5 a + 35 + \left(16 a + 43\right)\cdot 47 + \left(43 a + 42\right)\cdot 47^{2} + \left(28 a + 11\right)\cdot 47^{3} + \left(18 a + 16\right)\cdot 47^{4} + \left(a + 21\right)\cdot 47^{5} +O(47^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 16 + 12\cdot 47 + 22\cdot 47^{2} + 20\cdot 47^{3} + 13\cdot 47^{4} + 47^{5} +O(47^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 24 + 12\cdot 47 + 8\cdot 47^{2} + 44\cdot 47^{4} + 17\cdot 47^{5} +O(47^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 42 a + 45 + \left(30 a + 23\right)\cdot 47 + \left(3 a + 19\right)\cdot 47^{2} + \left(18 a + 26\right)\cdot 47^{3} + \left(28 a + 24\right)\cdot 47^{4} + \left(45 a + 5\right)\cdot 47^{5} +O(47^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 43 a + 38 + \left(28 a + 16\right)\cdot 47 + \left(44 a + 17\right)\cdot 47^{2} + \left(32 a + 30\right)\cdot 47^{3} + \left(40 a + 20\right)\cdot 47^{4} + \left(12 a + 31\right)\cdot 47^{5} +O(47^{6})\)
| |
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 |
$1$ | $1$ | $()$ | $9$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$40$ | $3$ | $(1,2,3)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
$72$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$72$ | $5$ | $(1,3,4,5,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.