Properties

Label 1.2e5.8t1.1c4
Dimension 1
Group $C_8$
Conductor $ 2^{5}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_8$
Conductor:$32= 2^{5} $
Artin number field: Splitting field of $f= x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 2 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_8$
Parity: Even
Corresponding Dirichlet character: \(\chi_{32}(5,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 4 + 3\cdot 31 + 3\cdot 31^{2} + 11\cdot 31^{3} + 22\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 + 14\cdot 31 + 4\cdot 31^{2} + 8\cdot 31^{3} + 22\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 10 + 18\cdot 31 + 29\cdot 31^{3} + 19\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 11 + 27\cdot 31 + 24\cdot 31^{2} + 11\cdot 31^{3} + 17\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 20 + 3\cdot 31 + 6\cdot 31^{2} + 19\cdot 31^{3} + 13\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 21 + 12\cdot 31 + 30\cdot 31^{2} + 31^{3} + 11\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 22 + 16\cdot 31 + 26\cdot 31^{2} + 22\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 27 + 27\cdot 31 + 27\cdot 31^{2} + 19\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

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$$4$$(1,7,8,2)(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.