Artin representation 1.247.6t1.b.a

• Label: 1.247.6t1.b.a
• Dimension: $1$
• Group: $C_6$
• Conductor: $247$
• Root number: not computed
• Indicator: $0$

Basic invariants

Dimension:	$1$
Group:	$C_6$
Conductor:	$247$$\medspace = 13 \cdot 19$
Artin field:	Galois closure of 6.6.286315237.1
Galois orbit size:	$2$
Smallest permutation container:	$C_6$
Parity:	even
Dirichlet character:	$\chi_{247}(64,\cdot)$
Projective image:	$C_1$
Projective field:	Galois closure of $\Q$

Defining polynomial

$f(x)$

$=$

$x^{6} - x^{5} - 22x^{4} + 5x^{3} + 73x^{2} - 58x + 1$ .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $x^{2} + 7x + 2$

Roots:

$r_{ 1 }$	$=$	$6 a + 6 + \left(4 a + 9\right)\cdot 11 + 10\cdot 11^{2} + 5\cdot 11^{3} + 6 a\cdot 11^{4} +O(11^{5})$
$r_{ 2 }$	$=$	$5 a + 8 + \left(6 a + 10\right)\cdot 11 + \left(10 a + 7\right)\cdot 11^{2} + \left(10 a + 5\right)\cdot 11^{3} + \left(4 a + 2\right)\cdot 11^{4} +O(11^{5})$
$r_{ 3 }$	$=$	$6 a + 8 + \left(4 a + 1\right)\cdot 11 + 9\cdot 11^{2} + 11^{3} + \left(6 a + 8\right)\cdot 11^{4} +O(11^{5})$
$r_{ 4 }$	$=$	$5 a + 10 + \left(6 a + 2\right)\cdot 11 + \left(10 a + 6\right)\cdot 11^{2} + \left(10 a + 1\right)\cdot 11^{3} + \left(4 a + 10\right)\cdot 11^{4} +O(11^{5})$
$r_{ 5 }$	$=$	$6 a + \left(4 a + 9\right)\cdot 11 + 9\cdot 11^{3} + \left(6 a + 4\right)\cdot 11^{4} +O(11^{5})$
$r_{ 6 }$	$=$	$5 a + 2 + \left(6 a + 10\right)\cdot 11 + \left(10 a + 8\right)\cdot 11^{2} + \left(10 a + 8\right)\cdot 11^{3} + \left(4 a + 6\right)\cdot 11^{4} +O(11^{5})$

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

Cycle notation

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

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

Character values on conjugacy classes

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