Properties

Label 2.273.8t17.d.b
Dimension $2$
Group $C_4\wr C_2$
Conductor $273$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \)
Artin stem field: 8.0.8719893.2
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.273.4t1.a.b
Projective image: $D_4$
Projective stem field: 4.2.322959.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 7 + 97\cdot 103 + 86\cdot 103^{2} + 66\cdot 103^{3} + 43\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 51\cdot 103 + 32\cdot 103^{2} + 35\cdot 103^{3} + 79\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 18 + 34\cdot 103 + 62\cdot 103^{2} + 66\cdot 103^{3} + 55\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 19 + 51\cdot 103 + 83\cdot 103^{2} + 89\cdot 103^{3} + 43\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 41 + 98\cdot 103 + 68\cdot 103^{2} + 28\cdot 103^{3} + 100\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 44 + 2\cdot 103 + 14\cdot 103^{2} + 22\cdot 103^{3} + 6\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 81 + 38\cdot 103 + 102\cdot 103^{2} + 80\cdot 103^{3} + 17\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 92 + 38\cdot 103 + 64\cdot 103^{2} + 21\cdot 103^{3} + 65\cdot 103^{4} +O(103^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.