Properties

Label 1.160.8t1.c.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: 8.8.33554432000000.2
Galois orbit size: $4$
Smallest permutation container: $C_8$
Parity: even
Dirichlet character: \(\chi_{160}(43,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{8} - 40 x^{6} + 500 x^{4} - 2000 x^{2} + 2450\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 2 + 34\cdot 73 + 14\cdot 73^{2} + 21\cdot 73^{3} + 8\cdot 73^{4} + 18\cdot 73^{5} +O(73^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 4 + 56\cdot 73 + 3\cdot 73^{2} + 40\cdot 73^{3} + 41\cdot 73^{4} + 10\cdot 73^{5} +O(73^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 23 + 10\cdot 73 + 4\cdot 73^{2} + 20\cdot 73^{3} + 13\cdot 73^{4} + 66\cdot 73^{5} +O(73^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 32 + 21\cdot 73 + 59\cdot 73^{2} + 4\cdot 73^{3} + 21\cdot 73^{4} + 57\cdot 73^{5} +O(73^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 41 + 51\cdot 73 + 13\cdot 73^{2} + 68\cdot 73^{3} + 51\cdot 73^{4} + 15\cdot 73^{5} +O(73^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 50 + 62\cdot 73 + 68\cdot 73^{2} + 52\cdot 73^{3} + 59\cdot 73^{4} + 6\cdot 73^{5} +O(73^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 69 + 16\cdot 73 + 69\cdot 73^{2} + 32\cdot 73^{3} + 31\cdot 73^{4} + 62\cdot 73^{5} +O(73^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 71 + 38\cdot 73 + 58\cdot 73^{2} + 51\cdot 73^{3} + 64\cdot 73^{4} + 54\cdot 73^{5} +O(73^{6})\)  Toggle raw display

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

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

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$$8$$(1,3,7,5,8,6,2,4)$$\zeta_{8}^{3}$
$1$$8$$(1,5,2,3,8,4,7,6)$$\zeta_{8}$
$1$$8$$(1,6,7,4,8,3,2,5)$$-\zeta_{8}^{3}$
$1$$8$$(1,4,2,6,8,5,7,3)$$-\zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.