Properties

Label 1.89.8t1.1c3
Dimension 1
Group $C_8$
Conductor $ 89 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_8$
Conductor:$89 $
Artin number field: Splitting field of $f= x^{8} - x^{7} + 6 x^{6} - 46 x^{5} - 143 x^{4} + 575 x^{3} + 1160 x^{2} - 16 x + 512 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_8$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{89}(77,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 3 + 41\cdot 97 + 84\cdot 97^{2} + 84\cdot 97^{3} + 29\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 31 + 45\cdot 97 + 20\cdot 97^{2} + 76\cdot 97^{3} + 81\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 60 + 17\cdot 97 + 84\cdot 97^{2} + 62\cdot 97^{3} + 4\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 67 + 6\cdot 97 + 69\cdot 97^{2} + 12\cdot 97^{3} + 38\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 68 + 33\cdot 97 + 44\cdot 97^{2} + 11\cdot 97^{3} + 26\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 81 + 33\cdot 97 + 56\cdot 97^{2} + 19\cdot 97^{3} + 95\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 83 + 8\cdot 97 + 73\cdot 97^{2} + 7\cdot 97^{3} + 37\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 93 + 6\cdot 97 + 53\cdot 97^{2} + 15\cdot 97^{3} + 75\cdot 97^{4} +O\left(97^{ 5 }\right)$

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

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

Character values on conjugacy classes

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