Properties

Label 1.5_17.8t1.2c2
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} + 10 x^{6} - 79 x^{5} + 134 x^{4} + 41 x^{3} + 245 x^{2} - 846 x + 596 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_8$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{85}(43,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 7 + 39\cdot 83 + 3\cdot 83^{2} + 39\cdot 83^{3} + 72\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 15 + 71\cdot 83 + 51\cdot 83^{2} + 45\cdot 83^{3} + 21\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 19 + 56\cdot 83 + 63\cdot 83^{2} + 34\cdot 83^{3} + 53\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 42 + 24\cdot 83 + 21\cdot 83^{2} + 28\cdot 83^{3} + 81\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 44 + 37\cdot 83 + 37\cdot 83^{2} + 13\cdot 83^{3} + 36\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 52 + 67\cdot 83 + 45\cdot 83^{2} + 38\cdot 83^{3} + 14\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 75 + 50\cdot 83 + 73\cdot 83^{2} + 25\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 79 + 67\cdot 83 + 34\cdot 83^{2} + 48\cdot 83^{3} + 27\cdot 83^{4} +O\left(83^{ 5 }\right)$

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

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

Character values on conjugacy classes

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