Artin representation 1.105.6t1.a.b

• Label: 1.105.6t1.a.b
• Dimension: $1$
• Group: $C_6$
• Conductor: $105$
• Root number: not computed
• Indicator: $0$

Basic invariants

Dimension:	$1$
Group:	$C_6$
Conductor:	$105$$\medspace = 3 \cdot 5 \cdot 7$
Artin field:	Galois closure of 6.0.8103375.1
Galois orbit size:	$2$
Smallest permutation container:	$C_6$
Parity:	odd
Dirichlet character:	$\chi_{105}(44,\cdot)$
Projective image:	$C_1$
Projective field:	Galois closure of $\Q$

Defining polynomial

$f(x)$

$=$

$x^{6} - x^{5} + 7x^{4} - 5x^{3} + 49x^{2} - 9x + 139$ .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{2} + 12x + 2$

Roots:

$r_{ 1 }$	$=$	$2 a + \left(2 a + 6\right)\cdot 13 + \left(10 a + 11\right)\cdot 13^{2} + 9 a\cdot 13^{3} + \left(11 a + 10\right)\cdot 13^{4} +O(13^{5})$
$r_{ 2 }$	$=$	$11 a + 2 + \left(10 a + 6\right)\cdot 13 + \left(2 a + 6\right)\cdot 13^{2} + 3 a\cdot 13^{3} + \left(a + 12\right)\cdot 13^{4} +O(13^{5})$
$r_{ 3 }$	$=$	$2 a + 3 + \left(2 a + 4\right)\cdot 13 + \left(10 a + 10\right)\cdot 13^{2} + \left(9 a + 7\right)\cdot 13^{3} + 11 a\cdot 13^{4} +O(13^{5})$
$r_{ 4 }$	$=$	$11 a + 5 + \left(10 a + 4\right)\cdot 13 + \left(2 a + 5\right)\cdot 13^{2} + \left(3 a + 7\right)\cdot 13^{3} + \left(a + 2\right)\cdot 13^{4} +O(13^{5})$
$r_{ 5 }$	$=$	$2 a + 1 + \left(2 a + 9\right)\cdot 13 + \left(10 a + 11\right)\cdot 13^{2} + \left(9 a + 4\right)\cdot 13^{3} + \left(11 a + 12\right)\cdot 13^{4} +O(13^{5})$
$r_{ 6 }$	$=$	$11 a + 3 + \left(10 a + 9\right)\cdot 13 + \left(2 a + 6\right)\cdot 13^{2} + \left(3 a + 4\right)\cdot 13^{3} + \left(a + 1\right)\cdot 13^{4} +O(13^{5})$

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

Cycle notation

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character value	Complex conjugation
$1$	$1$	$()$	$1$
$1$	$2$	$(1,2)(3,4)(5,6)$	$-1$	✓
$1$	$3$	$(1,5,3)(2,6,4)$	$-\zeta_{3} - 1$
$1$	$3$	$(1,3,5)(2,4,6)$	$\zeta_{3}$
$1$	$6$	$(1,6,3,2,5,4)$	$\zeta_{3} + 1$
$1$	$6$	$(1,4,5,2,3,6)$	$-\zeta_{3}$