Properties

Label 1.96.8t1.a.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: Galois closure of 8.8.173946175488.1
Galois orbit size: $4$
Smallest permutation container: $C_8$
Parity: even
Dirichlet character: \(\chi_{96}(59,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 9 + 9\cdot 79 + 15\cdot 79^{2} + 58\cdot 79^{3} + 66\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 19 + 43\cdot 79 + 15\cdot 79^{2} + 69\cdot 79^{3} + 8\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 21 + 6\cdot 79 + 67\cdot 79^{2} + 46\cdot 79^{3} + 57\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 22 + 59\cdot 79 + 36\cdot 79^{2} + 39\cdot 79^{3} + 45\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 57 + 19\cdot 79 + 42\cdot 79^{2} + 39\cdot 79^{3} + 33\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 58 + 72\cdot 79 + 11\cdot 79^{2} + 32\cdot 79^{3} + 21\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 60 + 35\cdot 79 + 63\cdot 79^{2} + 9\cdot 79^{3} + 70\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 70 + 69\cdot 79 + 63\cdot 79^{2} + 20\cdot 79^{3} + 12\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.