Properties

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

Related objects

Learn more

Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \)
Artin stem field: 8.0.8719893.1
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.273.4t1.a.a
Projective image: $D_4$
Projective stem field: 4.2.322959.1

Defining polynomial

$f(x)$$=$\(x^{8} - 3 x^{7} - x^{6} + 11 x^{5} - 6 x^{4} - 10 x^{3} + 9 x^{2} - x + 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 3 + 22\cdot 181 + 119\cdot 181^{2} + 127\cdot 181^{3} + 146\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 109\cdot 181 + 174\cdot 181^{2} + 179\cdot 181^{3} + 64\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 36 + 155\cdot 181 + 70\cdot 181^{2} + 151\cdot 181^{3} + 124\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 42 + 145\cdot 181 + 181^{2} + 36\cdot 181^{3} + 118\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 67 + 89\cdot 181 + 109\cdot 181^{2} + 58\cdot 181^{3} + 121\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 119 + 103\cdot 181 + 47\cdot 181^{2} + 136\cdot 181^{3} + 85\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 127 + 82\cdot 181 + 52\cdot 181^{2} + 106\cdot 181^{3} + 157\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 142 + 16\cdot 181 + 148\cdot 181^{2} + 108\cdot 181^{3} + 85\cdot 181^{4} +O(181^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.