Properties

Label 2.165.8t17.c.a
Dimension $2$
Group $C_4\wr C_2$
Conductor $165$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(165\)\(\medspace = 3 \cdot 5 \cdot 11 \)
Artin stem field: 8.0.16471125.1
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.165.4t1.a.a
Projective image: D_4
Projective stem field: 4.2.12375.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 2 + 24\cdot 59 + 48\cdot 59^{2} + 5\cdot 59^{3} + 40\cdot 59^{4} + 32\cdot 59^{5} +O(59^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 16 + 35\cdot 59 + 21\cdot 59^{2} + 42\cdot 59^{3} + 18\cdot 59^{4} + 27\cdot 59^{5} +O(59^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 19 + 44\cdot 59 + 57\cdot 59^{2} + 7\cdot 59^{3} + 11\cdot 59^{4} + 37\cdot 59^{5} +O(59^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 28 + 3\cdot 59 + 29\cdot 59^{2} + 45\cdot 59^{3} + 19\cdot 59^{4} + 11\cdot 59^{5} +O(59^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 30 + 23\cdot 59 + 30\cdot 59^{2} + 28\cdot 59^{3} + 39\cdot 59^{4} + 57\cdot 59^{5} +O(59^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 42 + 36\cdot 59 + 24\cdot 59^{2} + 48\cdot 59^{3} + 11\cdot 59^{4} + 13\cdot 59^{5} +O(59^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 48 + 38\cdot 59 + 10\cdot 59^{2} + 54\cdot 59^{3} + 2\cdot 59^{4} + 39\cdot 59^{5} +O(59^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 52 + 29\cdot 59 + 13\cdot 59^{2} + 3\cdot 59^{3} + 33\cdot 59^{4} + 17\cdot 59^{5} +O(59^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.