Artin representation 4.5e4_11e2_29e2.8t16.4c1

• Label: 4.5e4_11e2_29e2.8t16.4c1
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 5^{4} \cdot 11^{2} \cdot 29^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$63600625= 5^{4} \cdot 11^{2} \cdot 29^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 33 x^{6} - 71 x^{5} + 370 x^{4} - 546 x^{3} + 1653 x^{2} - 1313 x + 2131 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_8:C_2):C_2$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 4 + 40\cdot 71 + 9\cdot 71^{2} + 66\cdot 71^{3} + 17\cdot 71^{4} + 17\cdot 71^{5} + 71^{6} + 66\cdot 71^{7} +O\left(71^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 21 + 12\cdot 71 + 56\cdot 71^{2} + 42\cdot 71^{3} + 67\cdot 71^{4} + 53\cdot 71^{5} + 51\cdot 71^{6} + 38\cdot 71^{7} +O\left(71^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 22 + 17\cdot 71 + 46\cdot 71^{2} + 29\cdot 71^{3} + 42\cdot 71^{4} + 53\cdot 71^{5} + 61\cdot 71^{6} + 39\cdot 71^{7} +O\left(71^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 23 + 19\cdot 71 + 28\cdot 71^{2} + 49\cdot 71^{3} + 51\cdot 71^{4} + 50\cdot 71^{5} + 30\cdot 71^{6} + 22\cdot 71^{7} +O\left(71^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 30 + 64\cdot 71 + 48\cdot 71^{2} + 59\cdot 71^{3} + 38\cdot 71^{4} + 32\cdot 71^{5} + 56\cdot 71^{6} + 23\cdot 71^{7} +O\left(71^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 32 + 19\cdot 71 + 4\cdot 71^{3} + 3\cdot 71^{4} + 26\cdot 71^{5} + 7\cdot 71^{6} + 64\cdot 71^{7} +O\left(71^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 37 + 6\cdot 71 + 19\cdot 71^{2} + 57\cdot 71^{3} + 66\cdot 71^{4} + 57\cdot 71^{5} + 70\cdot 71^{6} + 43\cdot 71^{7} +O\left(71^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 47 + 33\cdot 71 + 4\cdot 71^{2} + 46\cdot 71^{3} + 66\cdot 71^{4} + 62\cdot 71^{5} + 3\cdot 71^{6} + 56\cdot 71^{7} +O\left(71^{ 8 }\right)$

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

Cycle notation

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

$(4,6)(5,8)$

$(2,7)(5,8)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.