Properties

Label 2.2e10_3e2.8t7.3c1
Dimension 2
Group $C_8:C_2$
Conductor $ 2^{10} \cdot 3^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$C_8:C_2$
Conductor:$9216= 2^{10} \cdot 3^{2} $
Artin number field: Splitting field of $f= x^{8} - 24 x^{6} - 108 x^{4} + 162 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_8:C_2$
Parity: Odd
Determinant: 1.2e4.4t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 113 }$ to precision 9.
Roots:
$r_{ 1 }$ $=$ $ 30 + 103\cdot 113 + 63\cdot 113^{2} + 96\cdot 113^{3} + 113^{4} + 22\cdot 113^{5} + 99\cdot 113^{6} + 10\cdot 113^{7} + 43\cdot 113^{8} +O\left(113^{ 9 }\right)$
$r_{ 2 }$ $=$ $ 32 + 83\cdot 113^{2} + 109\cdot 113^{3} + 2\cdot 113^{4} + 83\cdot 113^{5} + 9\cdot 113^{6} + 16\cdot 113^{7} + 17\cdot 113^{8} +O\left(113^{ 9 }\right)$
$r_{ 3 }$ $=$ $ 39 + 30\cdot 113 + 104\cdot 113^{2} + 101\cdot 113^{3} + 101\cdot 113^{4} + 51\cdot 113^{5} + 61\cdot 113^{6} + 84\cdot 113^{7} + 21\cdot 113^{8} +O\left(113^{ 9 }\right)$
$r_{ 4 }$ $=$ $ 46 + 26\cdot 113 + 75\cdot 113^{2} + 10\cdot 113^{3} + 90\cdot 113^{4} + 111\cdot 113^{5} + 43\cdot 113^{6} + 55\cdot 113^{7} + 101\cdot 113^{8} +O\left(113^{ 9 }\right)$
$r_{ 5 }$ $=$ $ 67 + 86\cdot 113 + 37\cdot 113^{2} + 102\cdot 113^{3} + 22\cdot 113^{4} + 113^{5} + 69\cdot 113^{6} + 57\cdot 113^{7} + 11\cdot 113^{8} +O\left(113^{ 9 }\right)$
$r_{ 6 }$ $=$ $ 74 + 82\cdot 113 + 8\cdot 113^{2} + 11\cdot 113^{3} + 11\cdot 113^{4} + 61\cdot 113^{5} + 51\cdot 113^{6} + 28\cdot 113^{7} + 91\cdot 113^{8} +O\left(113^{ 9 }\right)$
$r_{ 7 }$ $=$ $ 81 + 112\cdot 113 + 29\cdot 113^{2} + 3\cdot 113^{3} + 110\cdot 113^{4} + 29\cdot 113^{5} + 103\cdot 113^{6} + 96\cdot 113^{7} + 95\cdot 113^{8} +O\left(113^{ 9 }\right)$
$r_{ 8 }$ $=$ $ 83 + 9\cdot 113 + 49\cdot 113^{2} + 16\cdot 113^{3} + 111\cdot 113^{4} + 90\cdot 113^{5} + 13\cdot 113^{6} + 102\cdot 113^{7} + 69\cdot 113^{8} +O\left(113^{ 9 }\right)$

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

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

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