Artin representation 3.2e5_11e2.6t11.1c1

• Label: 3.2e5_11e2.6t11.1c1
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 2^{5} \cdot 11^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$3872= 2^{5} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} - 2 x^{2} - 2 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4\times C_2$
Parity:	Even
Determinant:	1.2e3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $

Roots:

$r_{ 1 }$	$=$	$ 58 a + 6 + \left(51 a + 63\right)\cdot 71 + \left(38 a + 9\right)\cdot 71^{2} + \left(27 a + 22\right)\cdot 71^{3} + \left(50 a + 21\right)\cdot 71^{4} + \left(4 a + 60\right)\cdot 71^{5} + \left(2 a + 34\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 42 a + 61 + \left(8 a + 25\right)\cdot 71 + \left(63 a + 33\right)\cdot 71^{2} + 20 a\cdot 71^{3} + 5 a\cdot 71^{4} + \left(22 a + 29\right)\cdot 71^{5} + \left(64 a + 2\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 13 a + 51 + \left(19 a + 37\right)\cdot 71 + \left(32 a + 35\right)\cdot 71^{2} + \left(43 a + 38\right)\cdot 71^{3} + \left(20 a + 23\right)\cdot 71^{4} + \left(66 a + 19\right)\cdot 71^{5} + \left(68 a + 34\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 44 + 6\cdot 71 + 57\cdot 71^{2} + 65\cdot 71^{3} + 50\cdot 71^{4} + 52\cdot 71^{5} + 33\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 50 + 7\cdot 71 + 68\cdot 71^{2} + 35\cdot 71^{3} + 56\cdot 71^{4} + 54\cdot 71^{5} + 69\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 29 a + 3 + \left(62 a + 1\right)\cdot 71 + \left(7 a + 9\right)\cdot 71^{2} + \left(50 a + 50\right)\cdot 71^{3} + \left(65 a + 60\right)\cdot 71^{4} + \left(48 a + 67\right)\cdot 71^{5} + \left(6 a + 37\right)\cdot 71^{6} +O\left(71^{ 7 }\right)$

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

Cycle notation

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

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

$(4,5)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.