Properties

Label 1.2e5.8t1.2c2
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: Odd
Corresponding Dirichlet character: \(\chi_{32}(27,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 1 + 40\cdot 47 + 5\cdot 47^{2} + 20\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 4 + 5\cdot 47 + 41\cdot 47^{2} + 19\cdot 47^{3} + 16\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 + 13\cdot 47 + 8\cdot 47^{2} + 22\cdot 47^{3} + 20\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 18 + 25\cdot 47 + 45\cdot 47^{3} + 35\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 29 + 21\cdot 47 + 46\cdot 47^{2} + 47^{3} + 11\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 36 + 33\cdot 47 + 38\cdot 47^{2} + 24\cdot 47^{3} + 26\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 43 + 41\cdot 47 + 5\cdot 47^{2} + 27\cdot 47^{3} + 30\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 46 + 6\cdot 47 + 41\cdot 47^{2} + 46\cdot 47^{3} + 26\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

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

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