Properties

Label 1.160.8t1.b.b
Dimension $1$
Group $C_8$
Conductor $160$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_8$
Conductor: \(160\)\(\medspace = 2^{5} \cdot 5 \)
Artin field: 8.0.33554432000000.1
Galois orbit size: $4$
Smallest permutation container: $C_8$
Parity: odd
Dirichlet character: \(\chi_{160}(37,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{8} + 40 x^{6} + 500 x^{4} + 2000 x^{2} + 50\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 7.

Roots:
$r_{ 1 }$ $=$ \( 3 + 21\cdot 31 + 10\cdot 31^{2} + 22\cdot 31^{3} + 22\cdot 31^{4} + 17\cdot 31^{5} + 25\cdot 31^{6} +O(31^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 21\cdot 31 + 8\cdot 31^{2} + 14\cdot 31^{3} + 30\cdot 31^{4} + 26\cdot 31^{5} + 13\cdot 31^{6} +O(31^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 11 + 9\cdot 31 + 16\cdot 31^{2} + 26\cdot 31^{3} + 5\cdot 31^{4} + 28\cdot 31^{5} + 30\cdot 31^{6} +O(31^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 13 + 23\cdot 31 + 24\cdot 31^{2} + 25\cdot 31^{3} + 27\cdot 31^{4} + 24\cdot 31^{5} +O(31^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 18 + 7\cdot 31 + 6\cdot 31^{2} + 5\cdot 31^{3} + 3\cdot 31^{4} + 6\cdot 31^{5} + 30\cdot 31^{6} +O(31^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 20 + 21\cdot 31 + 14\cdot 31^{2} + 4\cdot 31^{3} + 25\cdot 31^{4} + 2\cdot 31^{5} +O(31^{7})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 23 + 9\cdot 31 + 22\cdot 31^{2} + 16\cdot 31^{3} + 4\cdot 31^{5} + 17\cdot 31^{6} +O(31^{7})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 28 + 9\cdot 31 + 20\cdot 31^{2} + 8\cdot 31^{3} + 8\cdot 31^{4} + 13\cdot 31^{5} + 5\cdot 31^{6} +O(31^{7})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-1$
$1$$4$$(1,7,8,2)(3,5,6,4)$$-\zeta_{8}^{2}$
$1$$4$$(1,2,8,7)(3,4,6,5)$$\zeta_{8}^{2}$
$1$$8$$(1,5,7,6,8,4,2,3)$$\zeta_{8}^{3}$
$1$$8$$(1,6,2,5,8,3,7,4)$$\zeta_{8}$
$1$$8$$(1,4,7,3,8,5,2,6)$$-\zeta_{8}^{3}$
$1$$8$$(1,3,2,4,8,6,7,5)$$-\zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.