Properties

Label 8.2e18_3e16.36t555.1
Dimension 8
Group $A_6$
Conductor $ 2^{18} \cdot 3^{16}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$8$
Group:$A_6$
Conductor:$11284439629824= 2^{18} \cdot 3^{16} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $A_6$
Parity: Even

Galois action

Roots of defining polynomial

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} + 63 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 65 a + 49 + \left(29 a + 24\right)\cdot 67 + \left(44 a + 10\right)\cdot 67^{2} + \left(25 a + 60\right)\cdot 67^{3} + \left(31 a + 52\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 a + 41 + \left(37 a + 12\right)\cdot 67 + \left(22 a + 24\right)\cdot 67^{2} + \left(41 a + 51\right)\cdot 67^{3} + \left(35 a + 18\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 + 58\cdot 67 + 31\cdot 67^{2} + 61\cdot 67^{3} + 47\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 47 + 66\cdot 67 + 45\cdot 67^{2} + 42\cdot 67^{3} + 62\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 5 a + 45 + \left(18 a + 52\right)\cdot 67 + \left(36 a + 47\right)\cdot 67^{2} + \left(16 a + 44\right)\cdot 67^{3} + \left(58 a + 1\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 62 a + 65 + \left(48 a + 52\right)\cdot 67 + \left(30 a + 40\right)\cdot 67^{2} + \left(50 a + 7\right)\cdot 67^{3} + \left(8 a + 17\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2,3)$
$(1,2)(3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $8$ $8$
$45$ $2$ $(1,2)(3,4)$ $0$ $0$
$40$ $3$ $(1,2,3)(4,5,6)$ $-1$ $-1$
$40$ $3$ $(1,2,3)$ $-1$ $-1$
$90$ $4$ $(1,2,3,4)(5,6)$ $0$ $0$
$72$ $5$ $(1,2,3,4,5)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$
$72$ $5$ $(1,3,4,5,2)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.