Galois orbit of Artin representations 3.7744.4t5.c

• Label: 3.7744.4t5.c
• Dimension: $3$
• Group: $S_4$
• Conductor: $7744$
• Indicator: $1$

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	\(7744\)\(\medspace = 2^{6} \cdot 11^{2} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.2.3748096.1
Galois orbit size:	$1$
Smallest permutation container:	$S_4$
Parity:	odd
Projective image:	$S_4$
Projective field:	Galois closure of 6.2.3748096.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \)

Roots:

$r_{ 1 }$	$=$	\( 22 a + 20 + \left(4 a + 30\right)\cdot 43 + \left(29 a + 26\right)\cdot 43^{2} + 9\cdot 43^{3} + \left(31 a + 28\right)\cdot 43^{4} + \left(33 a + 27\right)\cdot 43^{5} + \left(5 a + 7\right)\cdot 43^{6} + \left(3 a + 30\right)\cdot 43^{7} + 17 a\cdot 43^{8} +O(43^{9})\)
$r_{ 2 }$	$=$	\( 23 + 19\cdot 43 + 10\cdot 43^{2} + 11\cdot 43^{3} + 3\cdot 43^{4} + 42\cdot 43^{5} + 5\cdot 43^{6} + 18\cdot 43^{7} + 27\cdot 43^{8} +O(43^{9})\)
$r_{ 3 }$	$=$	\( 21 a + 42 + \left(38 a + 12\right)\cdot 43 + \left(13 a + 8\right)\cdot 43^{2} + \left(42 a + 24\right)\cdot 43^{3} + \left(11 a + 15\right)\cdot 43^{4} + \left(9 a + 30\right)\cdot 43^{5} + \left(37 a + 22\right)\cdot 43^{6} + \left(39 a + 27\right)\cdot 43^{7} + \left(25 a + 14\right)\cdot 43^{8} +O(43^{9})\)
$r_{ 4 }$	$=$	\( 21 a + 23 + \left(38 a + 12\right)\cdot 43 + \left(13 a + 16\right)\cdot 43^{2} + \left(42 a + 33\right)\cdot 43^{3} + \left(11 a + 14\right)\cdot 43^{4} + \left(9 a + 15\right)\cdot 43^{5} + \left(37 a + 35\right)\cdot 43^{6} + \left(39 a + 12\right)\cdot 43^{7} + \left(25 a + 42\right)\cdot 43^{8} +O(43^{9})\)
$r_{ 5 }$	$=$	\( 20 + 23\cdot 43 + 32\cdot 43^{2} + 31\cdot 43^{3} + 39\cdot 43^{4} + 37\cdot 43^{6} + 24\cdot 43^{7} + 15\cdot 43^{8} +O(43^{9})\)
$r_{ 6 }$	$=$	\( 22 a + 1 + \left(4 a + 30\right)\cdot 43 + \left(29 a + 34\right)\cdot 43^{2} + 18\cdot 43^{3} + \left(31 a + 27\right)\cdot 43^{4} + \left(33 a + 12\right)\cdot 43^{5} + \left(5 a + 20\right)\cdot 43^{6} + \left(3 a + 15\right)\cdot 43^{7} + \left(17 a + 28\right)\cdot 43^{8} +O(43^{9})\)

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

Cycle notation

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

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

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$
$1$	$1$	$()$	$3$
$3$	$2$	$(1,4)(2,5)$	$-1$
$6$	$2$	$(1,3)(4,6)$	$1$
$8$	$3$	$(1,6,5)(2,4,3)$	$0$
$6$	$4$	$(1,2,4,5)(3,6)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.