Properties

Label 2.145.8t17.a.a
Dimension $2$
Group $C_4\wr C_2$
Conductor $145$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(145\)\(\medspace = 5 \cdot 29 \)
Artin stem field: 8.4.15243125.1
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.145.4t1.a.a
Projective image: $D_4$
Projective stem field: 4.0.121945.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 10 + 35\cdot 139 + 67\cdot 139^{2} + 29\cdot 139^{3} + 61\cdot 139^{4} +O(139^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 46 + 33\cdot 139 + 70\cdot 139^{2} + 108\cdot 139^{3} + 130\cdot 139^{4} +O(139^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 47 + 91\cdot 139 + 134\cdot 139^{2} + 48\cdot 139^{3} + 84\cdot 139^{4} +O(139^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 50 + 39\cdot 139 + 67\cdot 139^{2} + 129\cdot 139^{3} + 105\cdot 139^{4} +O(139^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 51 + 11\cdot 139 + 3\cdot 139^{2} + 122\cdot 139^{3} + 53\cdot 139^{4} +O(139^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 56 + 69\cdot 139 + 27\cdot 139^{2} + 127\cdot 139^{3} + 122\cdot 139^{4} +O(139^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 62 + 62\cdot 139 + 133\cdot 139^{2} + 25\cdot 139^{3} + 12\cdot 139^{4} +O(139^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 96 + 74\cdot 139 + 52\cdot 139^{2} + 103\cdot 139^{3} + 123\cdot 139^{4} +O(139^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.