Properties

Label 2.2e10_7.8t7.4c2
Dimension 2
Group $C_8:C_2$
Conductor $ 2^{10} \cdot 7 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$C_8:C_2$
Conductor:$7168= 2^{10} \cdot 7 $
Artin number field: Splitting field of $f= x^{8} + 24 x^{6} + 180 x^{4} + 448 x^{2} + 98 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_8:C_2$
Parity: Even
Determinant: 1.2e4_7.4t1.1c2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 113 }$ to precision 8.
Roots: \[ \begin{aligned} r_{ 1 } &= 4 + 48\cdot 113 + 30\cdot 113^{2} + 22\cdot 113^{3} + 57\cdot 113^{4} + 84\cdot 113^{5} + 45\cdot 113^{6} + 3\cdot 113^{7} +O\left(113^{ 8 }\right) \\ r_{ 2 } &= 5 + 85\cdot 113 + 62\cdot 113^{2} + 48\cdot 113^{3} + 83\cdot 113^{4} + 8\cdot 113^{5} + 51\cdot 113^{6} + 78\cdot 113^{7} +O\left(113^{ 8 }\right) \\ r_{ 3 } &= 17 + 14\cdot 113 + 82\cdot 113^{3} + 6\cdot 113^{4} + 17\cdot 113^{5} + 57\cdot 113^{6} + 26\cdot 113^{7} +O\left(113^{ 8 }\right) \\ r_{ 4 } &= 18 + 103\cdot 113 + 44\cdot 113^{2} + 84\cdot 113^{3} + 41\cdot 113^{4} + 12\cdot 113^{5} + 51\cdot 113^{6} + 75\cdot 113^{7} +O\left(113^{ 8 }\right) \\ r_{ 5 } &= 95 + 9\cdot 113 + 68\cdot 113^{2} + 28\cdot 113^{3} + 71\cdot 113^{4} + 100\cdot 113^{5} + 61\cdot 113^{6} + 37\cdot 113^{7} +O\left(113^{ 8 }\right) \\ r_{ 6 } &= 96 + 98\cdot 113 + 112\cdot 113^{2} + 30\cdot 113^{3} + 106\cdot 113^{4} + 95\cdot 113^{5} + 55\cdot 113^{6} + 86\cdot 113^{7} +O\left(113^{ 8 }\right) \\ r_{ 7 } &= 108 + 27\cdot 113 + 50\cdot 113^{2} + 64\cdot 113^{3} + 29\cdot 113^{4} + 104\cdot 113^{5} + 61\cdot 113^{6} + 34\cdot 113^{7} +O\left(113^{ 8 }\right) \\ r_{ 8 } &= 109 + 64\cdot 113 + 82\cdot 113^{2} + 90\cdot 113^{3} + 55\cdot 113^{4} + 28\cdot 113^{5} + 67\cdot 113^{6} + 109\cdot 113^{7} +O\left(113^{ 8 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-2$
$2$$2$$(2,7)(4,5)$$0$
$1$$4$$(1,3,8,6)(2,4,7,5)$$-2 \zeta_{4}$
$1$$4$$(1,6,8,3)(2,5,7,4)$$2 \zeta_{4}$
$2$$4$$(1,6,8,3)(2,4,7,5)$$0$
$2$$8$$(1,5,3,2,8,4,6,7)$$0$
$2$$8$$(1,2,6,5,8,7,3,4)$$0$
$2$$8$$(1,2,3,4,8,7,6,5)$$0$
$2$$8$$(1,4,6,2,8,5,3,7)$$0$
The blue line marks the conjugacy class containing complex conjugation.