Basic invariants
Dimension: | $5$ |
Group: | $A_6$ |
Conductor: | \(6718464\)\(\medspace = 2^{10} \cdot 3^{8} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.6718464.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_6$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $A_6$ |
Projective stem field: | Galois closure of 6.2.6718464.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 3x^{4} - 12x^{3} - 9x^{2} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: \( x^{2} + 63x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 58 a + 3 + \left(38 a + 63\right)\cdot 67 + \left(5 a + 26\right)\cdot 67^{2} + \left(47 a + 2\right)\cdot 67^{3} + \left(42 a + 54\right)\cdot 67^{4} +O(67^{5})\)
$r_{ 2 }$ |
$=$ |
\( 9 a + 34 + \left(28 a + 26\right)\cdot 67 + \left(61 a + 10\right)\cdot 67^{2} + \left(19 a + 51\right)\cdot 67^{3} + \left(24 a + 43\right)\cdot 67^{4} +O(67^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 63 + 33\cdot 67 + 21\cdot 67^{2} + 54\cdot 67^{3} + 2\cdot 67^{4} +O(67^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 51 + 31\cdot 67 + 9\cdot 67^{2} + 27\cdot 67^{3} + 41\cdot 67^{4} +O(67^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 59 a + 41 + \left(45 a + 27\right)\cdot 67 + \left(54 a + 13\right)\cdot 67^{2} + \left(34 a + 24\right)\cdot 67^{3} + \left(53 a + 40\right)\cdot 67^{4} +O(67^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 8 a + 9 + \left(21 a + 18\right)\cdot 67 + \left(12 a + 52\right)\cdot 67^{2} + \left(32 a + 41\right)\cdot 67^{3} + \left(13 a + 18\right)\cdot 67^{4} +O(67^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $5$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$40$ | $3$ | $(1,2,3)$ | $2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.