Properties

Label 5.3e6_29e4.6t15.1
Dimension 5
Group $A_6$
Conductor $ 3^{6} \cdot 29^{4}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$515607849= 3^{6} \cdot 29^{4} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{3} - 3 x + 4 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_6$
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 17\cdot 19 + 19^{2} + 6\cdot 19^{3} + 10\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 15 a + 14 + \left(a + 18\right)\cdot 19 + \left(8 a + 6\right)\cdot 19^{2} + \left(16 a + 12\right)\cdot 19^{3} + \left(5 a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 3 a + 1 + \left(9 a + 14\right)\cdot 19 + \left(5 a + 9\right)\cdot 19^{2} + \left(5 a + 8\right)\cdot 19^{3} + \left(9 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 + 19^{3} + 8\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 16 a + 4 + \left(9 a + 1\right)\cdot 19 + \left(13 a + 6\right)\cdot 19^{2} + \left(13 a + 8\right)\cdot 19^{3} + \left(9 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 4 a + 10 + \left(17 a + 5\right)\cdot 19 + \left(10 a + 13\right)\cdot 19^{2} + \left(2 a + 1\right)\cdot 19^{3} + \left(13 a + 11\right)\cdot 19^{4} +O\left(19^{ 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
$c1$
$1$ $1$ $()$ $5$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $2$
$40$ $3$ $(1,2,3)$ $-1$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$72$ $5$ $(1,2,3,4,5)$ $0$
$72$ $5$ $(1,3,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.