Artin representation 3.2e4_5e3_7e2.6t11.1c1

• Label: 3.2e4_5e3_7e2.6t11.1c1
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 2^{4} \cdot 5^{3} \cdot 7^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$98000= 2^{4} \cdot 5^{3} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{6} - x^{4} - 2 x^{3} + x^{2} - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4\times C_2$
Parity:	Even
Determinant:	1.5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 11 a + 60 + \left(19 a + 9\right)\cdot 61 + \left(56 a + 44\right)\cdot 61^{2} + \left(45 a + 39\right)\cdot 61^{3} + \left(31 a + 10\right)\cdot 61^{4} + \left(52 a + 54\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 9 + 31\cdot 61 + 13\cdot 61^{2} + 7\cdot 61^{3} + 48\cdot 61^{4} + 43\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 27 + 15\cdot 61 + 16\cdot 61^{2} + 36\cdot 61^{3} + 13\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 55 a + 11 + \left(55 a + 23\right)\cdot 61 + \left(48 a + 17\right)\cdot 61^{2} + \left(13 a + 22\right)\cdot 61^{3} + \left(15 a + 32\right)\cdot 61^{4} + \left(22 a + 25\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 50 a + 10 + \left(41 a + 18\right)\cdot 61 + \left(4 a + 20\right)\cdot 61^{2} + \left(15 a + 29\right)\cdot 61^{3} + \left(29 a + 57\right)\cdot 61^{4} + \left(8 a + 13\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 6 a + 5 + \left(5 a + 24\right)\cdot 61 + \left(12 a + 10\right)\cdot 61^{2} + \left(47 a + 48\right)\cdot 61^{3} + \left(45 a + 33\right)\cdot 61^{4} + \left(38 a + 32\right)\cdot 61^{5} +O\left(61^{ 6 }\right)$

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

Cycle notation

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

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

$(2,3)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.