Properties

Label 1.32.8t1.b
Dimension $1$
Group $C_8$
Conductor $32$
Indicator $0$

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Basic invariants

Dimension:$1$
Group:$C_8$
Conductor:\(32\)\(\medspace = 2^{5} \)
Artin number field: Galois closure of 8.0.2147483648.1
Galois orbit size: $4$
Smallest permutation container: $C_8$
Parity: odd
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$ $c2$ $c3$ $c4$
$1$ $1$ $()$ $1$ $1$ $1$ $1$
$1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-1$ $-1$ $-1$ $-1$
$1$ $4$ $(1,5,8,4)(2,6,7,3)$ $\zeta_{8}^{2}$ $-\zeta_{8}^{2}$ $\zeta_{8}^{2}$ $-\zeta_{8}^{2}$
$1$ $4$ $(1,4,8,5)(2,3,7,6)$ $-\zeta_{8}^{2}$ $\zeta_{8}^{2}$ $-\zeta_{8}^{2}$ $\zeta_{8}^{2}$
$1$ $8$ $(1,7,5,3,8,2,4,6)$ $\zeta_{8}$ $\zeta_{8}^{3}$ $-\zeta_{8}$ $-\zeta_{8}^{3}$
$1$ $8$ $(1,3,4,7,8,6,5,2)$ $\zeta_{8}^{3}$ $\zeta_{8}$ $-\zeta_{8}^{3}$ $-\zeta_{8}$
$1$ $8$ $(1,2,5,6,8,7,4,3)$ $-\zeta_{8}$ $-\zeta_{8}^{3}$ $\zeta_{8}$ $\zeta_{8}^{3}$
$1$ $8$ $(1,6,4,2,8,3,5,7)$ $-\zeta_{8}^{3}$ $-\zeta_{8}$ $\zeta_{8}^{3}$ $\zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.