Properties

Label 2.2e6_3e2_5e2.8t7.6c2
Dimension 2
Group $C_8:C_2$
Conductor $ 2^{6} \cdot 3^{2} \cdot 5^{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:$14400= 2^{6} \cdot 3^{2} \cdot 5^{2} $
Artin number field: Splitting field of $f= x^{8} - 120 x^{4} - 360 x^{2} + 720 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_8:C_2$
Parity: Odd
Determinant: 1.5.4t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 7.
Roots:
$r_{ 1 }$ $=$ $ 32 + 133\cdot 181 + 121\cdot 181^{2} + 88\cdot 181^{3} + 115\cdot 181^{4} + 42\cdot 181^{5} + 143\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 62 + 69\cdot 181 + 101\cdot 181^{2} + 17\cdot 181^{3} + 123\cdot 181^{4} + 45\cdot 181^{5} + 173\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 67 + 113\cdot 181 + 139\cdot 181^{2} + 57\cdot 181^{3} + 180\cdot 181^{4} + 125\cdot 181^{5} + 103\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 69 + 170\cdot 181 + 134\cdot 181^{2} + 58\cdot 181^{3} + 178\cdot 181^{4} + 76\cdot 181^{5} + 137\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 112 + 10\cdot 181 + 46\cdot 181^{2} + 122\cdot 181^{3} + 2\cdot 181^{4} + 104\cdot 181^{5} + 43\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 114 + 67\cdot 181 + 41\cdot 181^{2} + 123\cdot 181^{3} + 55\cdot 181^{5} + 77\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 119 + 111\cdot 181 + 79\cdot 181^{2} + 163\cdot 181^{3} + 57\cdot 181^{4} + 135\cdot 181^{5} + 7\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 149 + 47\cdot 181 + 59\cdot 181^{2} + 92\cdot 181^{3} + 65\cdot 181^{4} + 138\cdot 181^{5} + 37\cdot 181^{6} +O\left(181^{ 7 }\right)$

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

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

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