Properties

Label 2.136.8t17.a.b
Dimension $2$
Group $C_4\wr C_2$
Conductor $136$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(136\)\(\medspace = 2^{3} \cdot 17 \)
Artin stem field: 8.0.20123648.1
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.136.4t1.b.a
Projective image: $D_4$
Projective stem field: 4.2.39304.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 307 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 38 + 166\cdot 307 + 69\cdot 307^{2} + 158\cdot 307^{3} + 143\cdot 307^{4} +O(307^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 50 + 75\cdot 307 + 241\cdot 307^{2} + 13\cdot 307^{3} + 63\cdot 307^{4} +O(307^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 75 + 64\cdot 307 + 64\cdot 307^{2} + 257\cdot 307^{3} + 69\cdot 307^{4} +O(307^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 84 + 92\cdot 307 + 186\cdot 307^{2} + 31\cdot 307^{3} + 223\cdot 307^{4} +O(307^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 95 + 238\cdot 307 + 137\cdot 307^{2} + 64\cdot 307^{3} + 92\cdot 307^{4} +O(307^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 148 + 267\cdot 307 + 168\cdot 307^{2} + 146\cdot 307^{3} + 65\cdot 307^{4} +O(307^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 196 + 34\cdot 307 + 212\cdot 307^{2} + 262\cdot 307^{3} + 29\cdot 307^{4} +O(307^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 237 + 289\cdot 307 + 147\cdot 307^{2} + 293\cdot 307^{3} + 233\cdot 307^{4} +O(307^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.