Properties

Label 2.576.8t11.c.b
Dimension $2$
Group $Q_8:C_2$
Conductor $576$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $Q_8:C_2$
Conductor: \(576\)\(\medspace = 2^{6} \cdot 3^{2}\)
Artin stem field: 8.0.47775744.2
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Determinant: 1.8.2t1.b.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{2}, \sqrt{-3})\)

Defining polynomial

$f(x)$$=$\(x^{8} - 6 x^{6} + 18 x^{4} - 18 x^{2} + 9\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 2 + 89\cdot 97 + 32\cdot 97^{2} + 67\cdot 97^{3} + 83\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 3 + 32\cdot 97 + 43\cdot 97^{2} + 82\cdot 97^{3} + 78\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 5 + 3\cdot 97 + 39\cdot 97^{2} + 31\cdot 97^{3} + 40\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 29 + 25\cdot 97 + 82\cdot 97^{2} + 6\cdot 97^{3} + 86\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 68 + 71\cdot 97 + 14\cdot 97^{2} + 90\cdot 97^{3} + 10\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 92 + 93\cdot 97 + 57\cdot 97^{2} + 65\cdot 97^{3} + 56\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 94 + 64\cdot 97 + 53\cdot 97^{2} + 14\cdot 97^{3} + 18\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 95 + 7\cdot 97 + 64\cdot 97^{2} + 29\cdot 97^{3} + 13\cdot 97^{4} +O(97^{5})\)  Toggle raw display

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

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

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$$2$$(1,7)(2,8)(3,4)(5,6)$$0$
$2$$2$$(1,4)(2,3)(5,8)(6,7)$$0$
$2$$2$$(2,7)(4,5)$$0$
$1$$4$$(1,6,8,3)(2,4,7,5)$$2 \zeta_{4}$
$1$$4$$(1,3,8,6)(2,5,7,4)$$-2 \zeta_{4}$
$2$$4$$(1,2,8,7)(3,5,6,4)$$0$
$2$$4$$(1,6,8,3)(2,5,7,4)$$0$
$2$$4$$(1,5,8,4)(2,3,7,6)$$0$

The blue line marks the conjugacy class containing complex conjugation.