Properties

Label 2.165.8t17.d.a
Dimension $2$
Group $C_4\wr C_2$
Conductor $165$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(165\)\(\medspace = 3 \cdot 5 \cdot 11 \)
Artin stem field: 8.0.16471125.2
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.165.4t1.a.a
Projective image: D_4
Projective stem field: 4.2.12375.1

Defining polynomial

$f(x)$$=$\(x^{8} - x^{7} + x^{6} + 2 x^{5} - 3 x^{4} - 2 x^{3} + x^{2} + x + 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 52 + 39\cdot 269 + 148\cdot 269^{2} + 164\cdot 269^{3} + 60\cdot 269^{4} +O(269^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 60 + 178\cdot 269 + 199\cdot 269^{2} + 152\cdot 269^{3} + 180\cdot 269^{4} +O(269^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 70 + 90\cdot 269 + 188\cdot 269^{2} + 123\cdot 269^{3} + 265\cdot 269^{4} +O(269^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 73 + 25\cdot 269 + 98\cdot 269^{2} + 200\cdot 269^{3} + 195\cdot 269^{4} +O(269^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 128 + 88\cdot 269 + 56\cdot 269^{2} + 41\cdot 269^{3} + 179\cdot 269^{4} +O(269^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 130 + 246\cdot 269 + 231\cdot 269^{2} + 258\cdot 269^{3} + 263\cdot 269^{4} +O(269^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 145 + 70\cdot 269 + 16\cdot 269^{2} + 109\cdot 269^{3} + 186\cdot 269^{4} +O(269^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 150 + 68\cdot 269 + 137\cdot 269^{2} + 25\cdot 269^{3} + 13\cdot 269^{4} +O(269^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,6)(3,4)(5,7)$$-2$
$2$$2$$(2,6)(3,4)$$0$
$4$$2$$(1,3)(2,7)(4,8)(5,6)$$0$
$1$$4$$(1,7,8,5)(2,4,6,3)$$2 \zeta_{4}$
$1$$4$$(1,5,8,7)(2,3,6,4)$$-2 \zeta_{4}$
$2$$4$$(2,4,6,3)$$\zeta_{4} + 1$
$2$$4$$(2,3,6,4)$$-\zeta_{4} + 1$
$2$$4$$(1,8)(2,3,6,4)(5,7)$$-\zeta_{4} - 1$
$2$$4$$(1,8)(2,4,6,3)(5,7)$$\zeta_{4} - 1$
$2$$4$$(1,5,8,7)(2,4,6,3)$$0$
$4$$4$$(1,4,8,3)(2,7,6,5)$$0$
$4$$8$$(1,2,7,4,8,6,5,3)$$0$
$4$$8$$(1,4,5,2,8,3,7,6)$$0$

The blue line marks the conjugacy class containing complex conjugation.