Properties

Label 1.5_17.8t1.3c1
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} + 6 x^{5} + 49 x^{4} - 129 x^{3} + 500 x^{2} + 2044 x + 1616 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_8$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{85}(42,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 4 + 13\cdot 43^{2} + 42\cdot 43^{3} + 39\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 13 + 14\cdot 43 + 22\cdot 43^{2} + 22\cdot 43^{3} + 13\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 23 + 29\cdot 43 + 41\cdot 43^{2} + 41\cdot 43^{3} + 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 24 + 32\cdot 43 + 21\cdot 43^{2} + 43^{3} + 29\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 36 + 27\cdot 43 + 21\cdot 43^{2} + 11\cdot 43^{3} + 16\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 37 + 31\cdot 43 + 3\cdot 43^{2} + 33\cdot 43^{3} + 3\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 39 + 40\cdot 43 + 35\cdot 43^{2} + 43^{3} + 8\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 40 + 37\cdot 43 + 11\cdot 43^{2} + 17\cdot 43^{3} + 16\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

Character values on conjugacy classes

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