Artin representation 3.2e6_3e3_7e2_11e2.6t11.1c1

• Label: 3.2e6_3e3_7e2_11e2.6t11.1c1
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$10245312= 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{6} + 2 x^{4} + 15 x^{2} + 3 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4\times C_2$
Parity:	Odd
Determinant:	1.3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 9.

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

Roots:

$r_{ 1 }$	$=$	$ 57 a + 2 + \left(7 a + 55\right)\cdot 61 + \left(15 a + 26\right)\cdot 61^{2} + \left(3 a + 36\right)\cdot 61^{3} + \left(36 a + 44\right)\cdot 61^{4} + \left(26 a + 4\right)\cdot 61^{5} + \left(52 a + 48\right)\cdot 61^{6} + \left(57 a + 27\right)\cdot 61^{7} + \left(29 a + 44\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$
$r_{ 2 }$	$=$	$ 18 + 43\cdot 61 + 27\cdot 61^{2} + 13\cdot 61^{3} + 24\cdot 61^{4} + 30\cdot 61^{5} + 55\cdot 61^{6} + 34\cdot 61^{7} + 41\cdot 61^{8} +O\left(61^{ 9 }\right)$
$r_{ 3 }$	$=$	$ 39 a + 11 + \left(22 a + 8\right)\cdot 61 + \left(49 a + 17\right)\cdot 61^{2} + \left(20 a + 14\right)\cdot 61^{3} + \left(12 a + 4\right)\cdot 61^{4} + \left(5 a + 34\right)\cdot 61^{5} + \left(39 a + 13\right)\cdot 61^{6} + \left(23 a + 38\right)\cdot 61^{7} + \left(60 a + 42\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$
$r_{ 4 }$	$=$	$ 4 a + 59 + \left(53 a + 5\right)\cdot 61 + \left(45 a + 34\right)\cdot 61^{2} + \left(57 a + 24\right)\cdot 61^{3} + \left(24 a + 16\right)\cdot 61^{4} + \left(34 a + 56\right)\cdot 61^{5} + \left(8 a + 12\right)\cdot 61^{6} + \left(3 a + 33\right)\cdot 61^{7} + \left(31 a + 16\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$
$r_{ 5 }$	$=$	$ 43 + 17\cdot 61 + 33\cdot 61^{2} + 47\cdot 61^{3} + 36\cdot 61^{4} + 30\cdot 61^{5} + 5\cdot 61^{6} + 26\cdot 61^{7} + 19\cdot 61^{8} +O\left(61^{ 9 }\right)$
$r_{ 6 }$	$=$	$ 22 a + 50 + \left(38 a + 52\right)\cdot 61 + \left(11 a + 43\right)\cdot 61^{2} + \left(40 a + 46\right)\cdot 61^{3} + \left(48 a + 56\right)\cdot 61^{4} + \left(55 a + 26\right)\cdot 61^{5} + \left(21 a + 47\right)\cdot 61^{6} + \left(37 a + 22\right)\cdot 61^{7} + 18\cdot 61^{8} +O\left(61^{ 9 }\right)$

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

Cycle notation

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

$(3,6)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.