Properties

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

Related objects

Learn more

Basic invariants

Dimension: $1$
Group: $C_8$
Conductor: \(160\)\(\medspace = 2^{5} \cdot 5 \)
Artin field: 8.0.1342177280000.1
Galois orbit size: $4$
Smallest permutation container: $C_8$
Parity: odd
Dirichlet character: \(\chi_{160}(59,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{8} + 40 x^{6} + 500 x^{4} + 2000 x^{2} + 1250\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 2 + 48\cdot 79 + 22\cdot 79^{3} + 43\cdot 79^{4} + 75\cdot 79^{5} +O(79^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 16\cdot 79 + 67\cdot 79^{2} + 56\cdot 79^{3} + 41\cdot 79^{4} + 6\cdot 79^{5} +O(79^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 23 + 18\cdot 79 + 13\cdot 79^{2} + 54\cdot 79^{3} + 65\cdot 79^{4} + 47\cdot 79^{5} +O(79^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 24 + 9\cdot 79 + 60\cdot 79^{2} + 5\cdot 79^{3} + 64\cdot 79^{4} + 71\cdot 79^{5} +O(79^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 55 + 69\cdot 79 + 18\cdot 79^{2} + 73\cdot 79^{3} + 14\cdot 79^{4} + 7\cdot 79^{5} +O(79^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 56 + 60\cdot 79 + 65\cdot 79^{2} + 24\cdot 79^{3} + 13\cdot 79^{4} + 31\cdot 79^{5} +O(79^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 73 + 62\cdot 79 + 11\cdot 79^{2} + 22\cdot 79^{3} + 37\cdot 79^{4} + 72\cdot 79^{5} +O(79^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 77 + 30\cdot 79 + 78\cdot 79^{2} + 56\cdot 79^{3} + 35\cdot 79^{4} + 3\cdot 79^{5} +O(79^{6})\)  Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.