Galois orbit of Artin representations 4.23553.6t13.a

• Label: 4.23553.6t13.a
• Dimension: $4$
• Group: $C_3^2:D_4$
• Conductor: $23553$
• Indicator: $1$

Basic invariants

Dimension:	$4$
Group:	$C_3^2:D_4$
Conductor:	\(23553\)\(\medspace = 3^{2} \cdot 2617 \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.4.53769057339.1
Galois orbit size:	$1$
Smallest permutation container:	$C_3^2:D_4$
Parity:	even
Projective image:	$\SOPlus(4,2)$
Projective field:	Galois closure of 6.4.53769057339.1

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 $\mathbb{Q}_{ 67 }$: \( x^{2} + 63x + 2 \)

Roots:

$r_{ 1 }$	$=$	\( 27 a + 33 + \left(27 a + 6\right)\cdot 67 + \left(18 a + 26\right)\cdot 67^{2} + \left(8 a + 30\right)\cdot 67^{3} + \left(25 a + 38\right)\cdot 67^{4} +O(67^{5})\)
$r_{ 2 }$	$=$	\( 40 a + 7 + \left(39 a + 22\right)\cdot 67 + \left(48 a + 5\right)\cdot 67^{2} + \left(58 a + 45\right)\cdot 67^{3} + \left(41 a + 63\right)\cdot 67^{4} +O(67^{5})\)
$r_{ 3 }$	$=$	\( 65 a + 58 + \left(4 a + 21\right)\cdot 67 + \left(14 a + 33\right)\cdot 67^{2} + \left(18 a + 66\right)\cdot 67^{3} + \left(17 a + 16\right)\cdot 67^{4} +O(67^{5})\)
$r_{ 4 }$	$=$	\( 28 + 38\cdot 67 + 35\cdot 67^{2} + 58\cdot 67^{3} + 31\cdot 67^{4} +O(67^{5})\)
$r_{ 5 }$	$=$	\( 2 a + 50 + \left(62 a + 43\right)\cdot 67 + \left(52 a + 17\right)\cdot 67^{2} + \left(48 a + 58\right)\cdot 67^{3} + 49 a\cdot 67^{4} +O(67^{5})\)
$r_{ 6 }$	$=$	\( 27 + 67 + 16\cdot 67^{2} + 9\cdot 67^{3} + 49\cdot 67^{4} +O(67^{5})\)

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

Cycle notation

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

$(3,5,6)$

$(3,5)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$6$	$2$	$(1,3)(2,5)(4,6)$	$2$
$6$	$2$	$(2,4)$	$0$
$9$	$2$	$(2,4)(5,6)$	$0$
$4$	$3$	$(3,5,6)$	$-2$
$4$	$3$	$(1,2,4)(3,5,6)$	$1$
$18$	$4$	$(1,3)(2,6,4,5)$	$0$
$12$	$6$	$(1,3,2,5,4,6)$	$-1$
$12$	$6$	$(2,4)(3,5,6)$	$0$

The blue line marks the conjugacy class containing complex conjugation.