Properties

Label 2.117.8t17.b.b
Dimension $2$
Group $C_4\wr C_2$
Conductor $117$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(117\)\(\medspace = 3^{2} \cdot 13 \)
Artin stem field: Galois closure of 8.0.1601613.1
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.13.4t1.a.b
Projective image: $D_4$
Projective stem field: Galois closure of 4.2.6591.1

Defining polynomial

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

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

Roots:
$r_{ 1 }$ $=$ \( 48 + 52\cdot 139 + 37\cdot 139^{2} + 86\cdot 139^{3} + 30\cdot 139^{4} +O(139^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 55 + 24\cdot 139 + 70\cdot 139^{2} + 32\cdot 139^{3} + 93\cdot 139^{4} +O(139^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 74 + 93\cdot 139 + 61\cdot 139^{2} + 116\cdot 139^{3} + 51\cdot 139^{4} +O(139^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 77 + 83\cdot 139 + 15\cdot 139^{2} + 95\cdot 139^{3} + 98\cdot 139^{4} +O(139^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 90 + 116\cdot 139 + 20\cdot 139^{2} + 25\cdot 139^{3} + 118\cdot 139^{4} +O(139^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 93 + 94\cdot 139 + 130\cdot 139^{2} + 28\cdot 139^{3} + 62\cdot 139^{4} +O(139^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 122 + 72\cdot 139 + 124\cdot 139^{2} + 31\cdot 139^{3} + 123\cdot 139^{4} +O(139^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 136 + 17\cdot 139 + 95\cdot 139^{2} + 117\cdot 139^{4} +O(139^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.