Properties

Label 2.768.8t11.c.b
Dimension $2$
Group $Q_8:C_2$
Conductor $768$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $Q_8:C_2$
Conductor: \(768\)\(\medspace = 2^{8} \cdot 3 \)
Artin stem field: 8.0.150994944.1
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-2}, \sqrt{3})\)

Defining polynomial

$f(x)$$=$\(x^{8} - 4 x^{6} + 8 x^{4} - 12 x^{2} + 9\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 14 + 55\cdot 73 + 69\cdot 73^{2} + 53\cdot 73^{3} + 25\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 15 + 67\cdot 73 + 48\cdot 73^{2} + 44\cdot 73^{3} + 57\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 16 + 34\cdot 73 + 19\cdot 73^{2} + 40\cdot 73^{3} + 70\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 35 + 46\cdot 73 + 67\cdot 73^{2} + 69\cdot 73^{3} + 37\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 38 + 26\cdot 73 + 5\cdot 73^{2} + 3\cdot 73^{3} + 35\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 57 + 38\cdot 73 + 53\cdot 73^{2} + 32\cdot 73^{3} + 2\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 58 + 5\cdot 73 + 24\cdot 73^{2} + 28\cdot 73^{3} + 15\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 59 + 17\cdot 73 + 3\cdot 73^{2} + 19\cdot 73^{3} + 47\cdot 73^{4} +O(73^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.