Galois orbit of Artin representations 4.2e2_5e2_7e2_11e2.6t9.1

• Label: 4.2e2_5e2_7e2_11e2.6t9.1
• Dimension: 4
• Group: $S_3^2$
• Conductor: $ 2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$S_3^2$
Conductor:	$592900= 2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{6} - 2 x^{5} + 3 x^{4} + 17 x^{3} - 18 x^{2} + 19 x - 6 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_3^2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $

Roots:

$r_{ 1 }$	$=$	$ 36 + 26\cdot 41 + 2\cdot 41^{2} + 31\cdot 41^{3} + 13\cdot 41^{4} + 23\cdot 41^{5} + 2\cdot 41^{6} +O\left(41^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 2 a + 3 + \left(18 a + 11\right)\cdot 41 + \left(26 a + 11\right)\cdot 41^{2} + \left(33 a + 33\right)\cdot 41^{3} + \left(12 a + 2\right)\cdot 41^{4} + \left(12 a + 34\right)\cdot 41^{5} + \left(19 a + 33\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 30 + 7\cdot 41 + 39\cdot 41^{2} + 22\cdot 41^{3} + 30\cdot 41^{4} + 30\cdot 41^{5} + 9\cdot 41^{6} +O\left(41^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 36 a + 31 + \left(11 a + 27\right)\cdot 41 + \left(8 a + 12\right)\cdot 41^{2} + \left(39 a + 32\right)\cdot 41^{3} + \left(10 a + 16\right)\cdot 41^{4} + \left(a + 12\right)\cdot 41^{5} + \left(34 a + 30\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 5 a + 16 + \left(29 a + 27\right)\cdot 41 + \left(32 a + 25\right)\cdot 41^{2} + \left(a + 18\right)\cdot 41^{3} + \left(30 a + 10\right)\cdot 41^{4} + \left(39 a + 5\right)\cdot 41^{5} + \left(6 a + 8\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 39 a + 9 + \left(22 a + 22\right)\cdot 41 + \left(14 a + 31\right)\cdot 41^{2} + \left(7 a + 25\right)\cdot 41^{3} + \left(28 a + 7\right)\cdot 41^{4} + \left(28 a + 17\right)\cdot 41^{5} + \left(21 a + 38\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$

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

Cycle notation

$(2,3)(4,5)$

$(2,6)(4,5)$

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

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

$(3,6)(4,5)$

$(1,4)(2,3)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$3$	$2$	$(1,3)(2,4)(5,6)$	$0$
$3$	$2$	$(1,2)(3,4)(5,6)$	$0$
$9$	$2$	$(1,4)(2,3)$	$0$
$2$	$3$	$(1,5,4)(2,6,3)$	$-2$
$2$	$3$	$(1,4,5)(2,6,3)$	$-2$
$4$	$3$	$(1,4,5)$	$1$
$6$	$6$	$(1,3,4,6,5,2)$	$0$
$6$	$6$	$(1,2,4,6,5,3)$	$0$

The blue line marks the conjugacy class containing complex conjugation.