Artin representation 3.15864289.42t37.a.b

• Label: 3.15864289.42t37.a.b
• Dimension: $3$
• Group: $\GL(3,2)$
• Conductor: $15864289$
• Root number: not computed
• Indicator: $0$

Basic invariants

Dimension:	$3$
Group:	$\GL(3,2)$
Conductor:	$15864289$$\medspace = 7^{2} \cdot 569^{2}$
Artin stem field:	Galois closure of 7.3.15864289.1
Galois orbit size:	$2$
Smallest permutation container:	$\PSL(2,7)$
Parity:	even
Determinant:	1.1.1t1.a.a
Projective image:	$\GL(3,2)$
Projective stem field:	Galois closure of 7.3.15864289.1

Defining polynomial

$f(x)$

$=$

$x^{7} - x^{6} + x^{5} - 4x^{4} + 4x^{3} - 5x^{2} + 2x + 1$ .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{3} + 2x + 11$

Roots:

$r_{ 1 }$	$=$	$4 a^{2} + 11 a + 3 + \left(4 a + 12\right)\cdot 13 + \left(8 a^{2} + 6 a + 6\right)\cdot 13^{2} + \left(3 a^{2} + 8 a + 5\right)\cdot 13^{3} + 10 a^{2} 13^{4} + \left(2 a^{2} + 11 a + 3\right)\cdot 13^{5} +O(13^{6})$
$r_{ 2 }$	$=$	$8 + 9\cdot 13 + 12\cdot 13^{2} + 8\cdot 13^{3} + 11\cdot 13^{4} +O(13^{6})$
$r_{ 3 }$	$=$	$10 a^{2} + 8 a + 11 + \left(4 a^{2} + 8 a\right)\cdot 13 + \left(6 a^{2} + a + 9\right)\cdot 13^{2} + \left(3 a^{2} + 10 a + 9\right)\cdot 13^{3} + \left(7 a^{2} + a\right)\cdot 13^{4} + \left(10 a^{2} + 3 a + 9\right)\cdot 13^{5} +O(13^{6})$
$r_{ 4 }$	$=$	$9 a + \left(2 a^{2} + 6 a + 5\right)\cdot 13 + \left(3 a^{2} + a + 12\right)\cdot 13^{2} + \left(3 a^{2} + 8 a + 4\right)\cdot 13^{3} + \left(3 a^{2} + 9\right)\cdot 13^{4} + \left(2 a^{2} + 4 a + 7\right)\cdot 13^{5} +O(13^{6})$
$r_{ 5 }$	$=$	$6 a^{2} + 11 a + 8 + \left(a^{2} + 9 a + 8\right)\cdot 13 + \left(2 a^{2} + 10 a + 6\right)\cdot 13^{2} + \left(7 a^{2} + 11 a + 1\right)\cdot 13^{3} + \left(4 a^{2} + 6 a + 11\right)\cdot 13^{4} + \left(4 a^{2} + 10 a + 1\right)\cdot 13^{5} +O(13^{6})$
$r_{ 6 }$	$=$	$12 a^{2} + 7 a + 5 + \left(7 a^{2} + 12 a + 9\right)\cdot 13 + \left(11 a^{2} + 4 a + 11\right)\cdot 13^{2} + \left(5 a^{2} + 7 a + 12\right)\cdot 13^{3} + \left(8 a^{2} + 10 a + 10\right)\cdot 13^{4} + \left(12 a^{2} + 11 a + 11\right)\cdot 13^{5} +O(13^{6})$
$r_{ 7 }$	$=$	$7 a^{2} + 6 a + 5 + \left(9 a^{2} + 9 a + 6\right)\cdot 13 + \left(7 a^{2} + 5\right)\cdot 13^{2} + \left(2 a^{2} + 6 a + 8\right)\cdot 13^{3} + \left(5 a^{2} + 5 a + 7\right)\cdot 13^{4} + \left(6 a^{2} + 11 a + 4\right)\cdot 13^{5} +O(13^{6})$

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation

$(2,6,7,4)(3,5)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 7 }$	Character value	Complex conjugation
$1$	$1$	$()$	$3$
$21$	$2$	$(1,4)(2,5)$	$-1$	✓
$56$	$3$	$(1,6,4)(2,5,7)$	$0$
$42$	$4$	$(1,5,6,7)(2,3)$	$1$
$24$	$7$	$(1,2,3,5,6,7,4)$	$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
$24$	$7$	$(1,5,4,3,7,2,6)$	$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$