Properties

Label 2.3e2_5e2_41.8t7.2c1
Dimension 2
Group $C_8:C_2$
Conductor $ 3^{2} \cdot 5^{2} \cdot 41 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$C_8:C_2$
Conductor:$9225= 3^{2} \cdot 5^{2} \cdot 41 $
Artin number field: Splitting field of $f= x^{8} - x^{7} + 2 x^{6} - 28 x^{5} - 35 x^{4} + 193 x^{3} - 133 x^{2} + x + 331 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_8:C_2$
Parity: Odd
Determinant: 1.5_41.4t1.2c2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 6 + 51\cdot 179 + 70\cdot 179^{2} + 85\cdot 179^{3} + 176\cdot 179^{4} +O\left(179^{ 5 }\right) \\ r_{ 2 } &= 28 + 131\cdot 179 + 77\cdot 179^{2} + 44\cdot 179^{3} + 68\cdot 179^{4} +O\left(179^{ 5 }\right) \\ r_{ 3 } &= 45 + 36\cdot 179 + 133\cdot 179^{2} + 145\cdot 179^{3} + 139\cdot 179^{4} +O\left(179^{ 5 }\right) \\ r_{ 4 } &= 70 + 18\cdot 179 + 111\cdot 179^{2} + 57\cdot 179^{3} + 67\cdot 179^{4} +O\left(179^{ 5 }\right) \\ r_{ 5 } &= 111 + 8\cdot 179^{2} + 126\cdot 179^{3} + 110\cdot 179^{4} +O\left(179^{ 5 }\right) \\ r_{ 6 } &= 146 + 47\cdot 179 + 83\cdot 179^{2} + 112\cdot 179^{3} + 60\cdot 179^{4} +O\left(179^{ 5 }\right) \\ r_{ 7 } &= 148 + 32\cdot 179 + 59\cdot 179^{2} + 142\cdot 179^{3} + 81\cdot 179^{4} +O\left(179^{ 5 }\right) \\ r_{ 8 } &= 163 + 39\cdot 179 + 173\cdot 179^{2} + 179^{3} + 11\cdot 179^{4} +O\left(179^{ 5 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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