Properties

Label 1.32.8t1.a.b
Dimension $1$
Group $C_8$
Conductor $32$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_8$
Conductor: \(32\)\(\medspace = 2^{5}\)
Artin field: Galois closure of \(\Q(\zeta_{32})^+\)
Galois orbit size: $4$
Smallest permutation container: $C_8$
Parity: even
Dirichlet character: \(\chi_{32}(21,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 4 + 3\cdot 31 + 3\cdot 31^{2} + 11\cdot 31^{3} + 22\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 + 14\cdot 31 + 4\cdot 31^{2} + 8\cdot 31^{3} + 22\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 10 + 18\cdot 31 + 29\cdot 31^{3} + 19\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 11 + 27\cdot 31 + 24\cdot 31^{2} + 11\cdot 31^{3} + 17\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 20 + 3\cdot 31 + 6\cdot 31^{2} + 19\cdot 31^{3} + 13\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 21 + 12\cdot 31 + 30\cdot 31^{2} + 31^{3} + 11\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 22 + 16\cdot 31 + 26\cdot 31^{2} + 22\cdot 31^{3} + 8\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 27 + 27\cdot 31 + 27\cdot 31^{2} + 19\cdot 31^{3} + 8\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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.