Properties

Label 2.3e2_5e2_7e2.8t7.2c2
Dimension 2
Group $C_8:C_2$
Conductor $ 3^{2} \cdot 5^{2} \cdot 7^{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:$11025= 3^{2} \cdot 5^{2} \cdot 7^{2} $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} - 2 x^{6} + 31 x^{5} - 65 x^{4} - 239 x^{3} + 58 x^{2} + 868 x + 961 $ 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 8.
Roots: \[ \begin{aligned} r_{ 1 } &= 52 + 73\cdot 181 + 23\cdot 181^{2} + 118\cdot 181^{3} + 98\cdot 181^{4} + 43\cdot 181^{5} + 33\cdot 181^{6} + 101\cdot 181^{7} +O\left(181^{ 8 }\right) \\ r_{ 2 } &= 58 + 2\cdot 181 + 113\cdot 181^{2} + 177\cdot 181^{3} + 170\cdot 181^{4} + 158\cdot 181^{5} + 102\cdot 181^{6} + 105\cdot 181^{7} +O\left(181^{ 8 }\right) \\ r_{ 3 } &= 69 + 95\cdot 181 + 116\cdot 181^{2} + 167\cdot 181^{3} + 77\cdot 181^{4} + 124\cdot 181^{5} + 95\cdot 181^{6} + 40\cdot 181^{7} +O\left(181^{ 8 }\right) \\ r_{ 4 } &= 80 + 178\cdot 181 + 45\cdot 181^{2} + 93\cdot 181^{3} + 127\cdot 181^{4} + 93\cdot 181^{5} + 25\cdot 181^{6} + 98\cdot 181^{7} +O\left(181^{ 8 }\right) \\ r_{ 5 } &= 90 + 174\cdot 181 + 47\cdot 181^{2} + 179\cdot 181^{3} + 63\cdot 181^{4} + 35\cdot 181^{5} + 86\cdot 181^{6} + 83\cdot 181^{7} +O\left(181^{ 8 }\right) \\ r_{ 6 } &= 121 + 72\cdot 181 + 135\cdot 181^{2} + 117\cdot 181^{3} + 96\cdot 181^{4} + 76\cdot 181^{5} + 77\cdot 181^{6} + 26\cdot 181^{7} +O\left(181^{ 8 }\right) \\ r_{ 7 } &= 124 + 94\cdot 181 + 151\cdot 181^{2} + 102\cdot 181^{3} + 92\cdot 181^{4} + 108\cdot 181^{5} + 154\cdot 181^{6} + 139\cdot 181^{7} +O\left(181^{ 8 }\right) \\ r_{ 8 } &= 132 + 32\cdot 181 + 90\cdot 181^{2} + 129\cdot 181^{3} + 176\cdot 181^{4} + 82\cdot 181^{5} + 148\cdot 181^{6} + 128\cdot 181^{7} +O\left(181^{ 8 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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