Artin representation 4.2e2_13e2_79e2.8t23.4c1

• Label: 4.2e2_13e2_79e2.8t23.4c1
• Dimension: 4
• Group: $\textrm{GL(2,3)}$
• Conductor: $ 2^{2} \cdot 13^{2} \cdot 79^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$\textrm{GL(2,3)}$
Conductor:	$4218916= 2^{2} \cdot 13^{2} \cdot 79^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 11 x^{6} - 19 x^{5} + 13 x^{4} + x^{3} + 4 x^{2} - 7 x - 2 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$\textrm{GL(2,3)}$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 26 a + 16 + \left(23 a + 32\right)\cdot 53 + \left(37 a + 33\right)\cdot 53^{2} + \left(26 a + 33\right)\cdot 53^{3} + \left(7 a + 18\right)\cdot 53^{4} + \left(28 a + 52\right)\cdot 53^{5} + \left(3 a + 10\right)\cdot 53^{6} + \left(a + 29\right)\cdot 53^{7} + \left(5 a + 46\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$
$r_{ 2 }$	$=$	$ 47 a + 39 + \left(47 a + 33\right)\cdot 53 + \left(14 a + 20\right)\cdot 53^{2} + \left(46 a + 47\right)\cdot 53^{3} + \left(18 a + 11\right)\cdot 53^{4} + \left(27 a + 34\right)\cdot 53^{5} + \left(23 a + 19\right)\cdot 53^{6} + \left(20 a + 50\right)\cdot 53^{7} + \left(48 a + 45\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$
$r_{ 3 }$	$=$	$ 6 a + 15 + \left(5 a + 19\right)\cdot 53 + \left(38 a + 32\right)\cdot 53^{2} + \left(6 a + 5\right)\cdot 53^{3} + \left(34 a + 41\right)\cdot 53^{4} + \left(25 a + 18\right)\cdot 53^{5} + \left(29 a + 33\right)\cdot 53^{6} + \left(32 a + 2\right)\cdot 53^{7} + \left(4 a + 7\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$
$r_{ 4 }$	$=$	$ 26 a + 40 + \left(23 a + 5\right)\cdot 53 + \left(37 a + 52\right)\cdot 53^{2} + \left(26 a + 2\right)\cdot 53^{3} + \left(7 a + 31\right)\cdot 53^{4} + \left(28 a + 1\right)\cdot 53^{5} + \left(3 a + 3\right)\cdot 53^{6} + \left(a + 23\right)\cdot 53^{7} + \left(5 a + 40\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$
$r_{ 5 }$	$=$	$ 27 a + 38 + \left(29 a + 20\right)\cdot 53 + \left(15 a + 19\right)\cdot 53^{2} + \left(26 a + 19\right)\cdot 53^{3} + \left(45 a + 34\right)\cdot 53^{4} + 24 a\cdot 53^{5} + \left(49 a + 42\right)\cdot 53^{6} + \left(51 a + 23\right)\cdot 53^{7} + \left(47 a + 6\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$
$r_{ 6 }$	$=$	$ 45 + 42\cdot 53 + 10\cdot 53^{2} + 24\cdot 53^{3} + 15\cdot 53^{4} + 8\cdot 53^{5} + 45\cdot 53^{6} + 36\cdot 53^{7} + 11\cdot 53^{8} +O\left(53^{ 9 }\right)$
$r_{ 7 }$	$=$	$ 9 + 10\cdot 53 + 42\cdot 53^{2} + 28\cdot 53^{3} + 37\cdot 53^{4} + 44\cdot 53^{5} + 7\cdot 53^{6} + 16\cdot 53^{7} + 41\cdot 53^{8} +O\left(53^{ 9 }\right)$
$r_{ 8 }$	$=$	$ 27 a + 14 + \left(29 a + 47\right)\cdot 53 + 15 a\cdot 53^{2} + \left(26 a + 50\right)\cdot 53^{3} + \left(45 a + 21\right)\cdot 53^{4} + \left(24 a + 51\right)\cdot 53^{5} + \left(49 a + 49\right)\cdot 53^{6} + \left(51 a + 29\right)\cdot 53^{7} + \left(47 a + 12\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation

$(2,8,6)(3,4,7)$

$(1,2,5,3)(4,7,8,6)$

$(1,5)(2,3)(4,8)(6,7)$

$(2,3)(4,6)(7,8)$

$(1,6,5,7)(2,8,3,4)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character value
$1$	$1$	$()$	$4$
$1$	$2$	$(1,5)(2,3)(4,8)(6,7)$	$-4$
$12$	$2$	$(2,3)(4,6)(7,8)$	$0$
$8$	$3$	$(1,2,4)(3,8,5)$	$1$
$6$	$4$	$(1,2,5,3)(4,7,8,6)$	$0$
$8$	$6$	$(1,8,2,5,4,3)(6,7)$	$-1$
$6$	$8$	$(1,6,2,4,5,7,3,8)$	$0$
$6$	$8$	$(1,7,2,8,5,6,3,4)$	$0$

The blue line marks the conjugacy class containing complex conjugation.