Properties

Label 1.5_17.8t1.1c1
Dimension 1
Group $C_8$
Conductor $ 5 \cdot 17 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_8$
Conductor:$85= 5 \cdot 17 $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 24 x^{6} + 23 x^{5} + 151 x^{4} - 197 x^{3} - 214 x^{2} + 429 x - 169 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_8$
Parity: Even
Corresponding Dirichlet character: \(\chi_{85}(59,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 20 + 45\cdot 101 + 52\cdot 101^{2} + 19\cdot 101^{3} + 26\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 21 + 49\cdot 101 + 84\cdot 101^{2} + 10\cdot 101^{3} + 75\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 32 + 101 + 101^{2} + 48\cdot 101^{3} + 55\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 35 + 23\cdot 101 + 29\cdot 101^{2} + 55\cdot 101^{3} + 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 46 + 80\cdot 101 + 45\cdot 101^{2} + 83\cdot 101^{3} + 7\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 70 + 45\cdot 101 + 89\cdot 101^{2} + 30\cdot 101^{3} + 44\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 87 + 35\cdot 101 + 54\cdot 101^{2} + 81\cdot 101^{3} + 14\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 94 + 21\cdot 101 + 47\cdot 101^{2} + 74\cdot 101^{3} + 77\cdot 101^{4} +O\left(101^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,2)(3,8)(4,6)(5,7)$$-1$
$1$$4$$(1,5,2,7)(3,4,8,6)$$\zeta_{8}^{2}$
$1$$4$$(1,7,2,5)(3,6,8,4)$$-\zeta_{8}^{2}$
$1$$8$$(1,4,5,8,2,6,7,3)$$\zeta_{8}$
$1$$8$$(1,8,7,4,2,3,5,6)$$\zeta_{8}^{3}$
$1$$8$$(1,6,5,3,2,4,7,8)$$-\zeta_{8}$
$1$$8$$(1,3,7,6,2,8,5,4)$$-\zeta_{8}^{3}$
The blue line marks the conjugacy class containing complex conjugation.