Properties

Label 1.96.8t1.b.c
Dimension $1$
Group $C_8$
Conductor $96$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_8$
Conductor: \(96\)\(\medspace = 2^{5} \cdot 3 \)
Artin field: 8.0.173946175488.1
Galois orbit size: $4$
Smallest permutation container: $C_8$
Parity: odd
Dirichlet character: \(\chi_{96}(53,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{8} + 24 x^{6} + 180 x^{4} + 432 x^{2} + 162\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 3 + 11\cdot 31 + 13\cdot 31^{2} + 9\cdot 31^{3} + 23\cdot 31^{4} + 21\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 6\cdot 31 + 13\cdot 31^{2} + 21\cdot 31^{3} + 30\cdot 31^{4} + 25\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 13 + 22\cdot 31 + 3\cdot 31^{2} + 20\cdot 31^{3} + 22\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 14 + 14\cdot 31 + 23\cdot 31^{2} + 19\cdot 31^{3} + 22\cdot 31^{4} + 24\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 17 + 16\cdot 31 + 7\cdot 31^{2} + 11\cdot 31^{3} + 8\cdot 31^{4} + 6\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 18 + 8\cdot 31 + 27\cdot 31^{2} + 10\cdot 31^{3} + 30\cdot 31^{4} + 8\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 25 + 24\cdot 31 + 17\cdot 31^{2} + 9\cdot 31^{3} + 5\cdot 31^{5} +O(31^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 28 + 19\cdot 31 + 17\cdot 31^{2} + 21\cdot 31^{3} + 7\cdot 31^{4} + 9\cdot 31^{5} +O(31^{6})\)  Toggle raw display

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

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

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)$$-\zeta_{8}$
$1$$8$$(1,3,4,7,8,6,5,2)$$-\zeta_{8}^{3}$
$1$$8$$(1,2,5,6,8,7,4,3)$$\zeta_{8}$
$1$$8$$(1,6,4,2,8,3,5,7)$$\zeta_{8}^{3}$

The blue line marks the conjugacy class containing complex conjugation.