Properties

Label 2.2880.8t11.e.a
Dimension $2$
Group $Q_8:C_2$
Conductor $2880$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $Q_8:C_2$
Conductor: \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \)
Artin stem field: 8.0.19110297600.5
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Determinant: 1.40.2t1.b.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{5}, \sqrt{-6})\)

Defining polynomial

$f(x)$$=$\(x^{8} - 4 x^{6} - 12 x^{5} + 18 x^{4} + 48 x^{3} - 16 x^{2} - 168 x + 166\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 15 + 29\cdot 59 + 23\cdot 59^{2} + 8\cdot 59^{3} +O(59^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 16 + 58\cdot 59 + 30\cdot 59^{2} + 52\cdot 59^{3} + 29\cdot 59^{4} +O(59^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 20 + 22\cdot 59 + 42\cdot 59^{2} + 36\cdot 59^{3} + 45\cdot 59^{4} +O(59^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 21 + 51\cdot 59 + 49\cdot 59^{2} + 21\cdot 59^{3} + 16\cdot 59^{4} +O(59^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 35 + 49\cdot 59^{2} + 13\cdot 59^{3} + 46\cdot 59^{4} +O(59^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 37 + 37\cdot 59 + 25\cdot 59^{2} + 23\cdot 59^{3} + 19\cdot 59^{4} +O(59^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 45 + 58\cdot 59 + 18\cdot 59^{2} + 5\cdot 59^{3} + 23\cdot 59^{4} +O(59^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 47 + 36\cdot 59 + 54\cdot 59^{2} + 14\cdot 59^{3} + 55\cdot 59^{4} +O(59^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.