Properties

Label 1.224.8t1.b.c
Dimension $1$
Group $C_8$
Conductor $224$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_8$
Conductor: \(224\)\(\medspace = 2^{5} \cdot 7 \)
Artin field: 8.0.5156108238848.1
Galois orbit size: $4$
Smallest permutation container: $C_8$
Parity: odd
Dirichlet character: \(\chi_{224}(69,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{8} + 56 x^{6} + 980 x^{4} + 5488 x^{2} + 4802\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 7.

Roots:
$r_{ 1 }$ $=$ \( 1 + 23\cdot 47 + 24\cdot 47^{2} + 10\cdot 47^{3} + 45\cdot 47^{4} + 30\cdot 47^{5} + 5\cdot 47^{6} +O(47^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 17 + 8\cdot 47 + 45\cdot 47^{2} + 23\cdot 47^{3} + 2\cdot 47^{4} + 9\cdot 47^{5} + 23\cdot 47^{6} +O(47^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 21 + 30\cdot 47 + 19\cdot 47^{2} + 2\cdot 47^{3} + 12\cdot 47^{4} + 23\cdot 47^{5} + 17\cdot 47^{6} +O(47^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 23 + 8\cdot 47 + 27\cdot 47^{2} + 22\cdot 47^{3} + 34\cdot 47^{4} + 20\cdot 47^{5} + 14\cdot 47^{6} +O(47^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 24 + 38\cdot 47 + 19\cdot 47^{2} + 24\cdot 47^{3} + 12\cdot 47^{4} + 26\cdot 47^{5} + 32\cdot 47^{6} +O(47^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 26 + 16\cdot 47 + 27\cdot 47^{2} + 44\cdot 47^{3} + 34\cdot 47^{4} + 23\cdot 47^{5} + 29\cdot 47^{6} +O(47^{7})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 30 + 38\cdot 47 + 47^{2} + 23\cdot 47^{3} + 44\cdot 47^{4} + 37\cdot 47^{5} + 23\cdot 47^{6} +O(47^{7})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 46 + 23\cdot 47 + 22\cdot 47^{2} + 36\cdot 47^{3} + 47^{4} + 16\cdot 47^{5} + 41\cdot 47^{6} +O(47^{7})\)  Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.