Properties

Label 2.2304.8t5.b.a
Dimension $2$
Group $Q_8$
Conductor $2304$
Root number $-1$
Indicator $-1$

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Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
Frobenius-Schur indicator: $-1$
Root number: $-1$
Artin field: Galois closure of 8.0.12230590464.1
Galois orbit size: $1$
Smallest permutation container: $Q_8$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{3})\)

Defining polynomial

$f(x)$$=$ \( x^{8} + 12x^{6} + 36x^{4} + 36x^{2} + 9 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 9.

Roots:
$r_{ 1 }$ $=$ \( 1 + 20\cdot 47 + 35\cdot 47^{2} + 41\cdot 47^{3} + 9\cdot 47^{4} + 42\cdot 47^{5} + 25\cdot 47^{6} + 13\cdot 47^{7} + 32\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 29\cdot 47 + 9\cdot 47^{2} + 6\cdot 47^{3} + 14\cdot 47^{4} + 5\cdot 47^{5} + 15\cdot 47^{6} + 37\cdot 47^{7} + 42\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 12 + 36\cdot 47 + 10\cdot 47^{2} + 29\cdot 47^{3} + 33\cdot 47^{4} + 26\cdot 47^{5} + 20\cdot 47^{6} + 7\cdot 47^{7} + 5\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 22 + 38\cdot 47 + 43\cdot 47^{2} + 35\cdot 47^{3} + 22\cdot 47^{4} + 17\cdot 47^{5} + 5\cdot 47^{6} + 44\cdot 47^{7} + 32\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 25 + 8\cdot 47 + 3\cdot 47^{2} + 11\cdot 47^{3} + 24\cdot 47^{4} + 29\cdot 47^{5} + 41\cdot 47^{6} + 2\cdot 47^{7} + 14\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 35 + 10\cdot 47 + 36\cdot 47^{2} + 17\cdot 47^{3} + 13\cdot 47^{4} + 20\cdot 47^{5} + 26\cdot 47^{6} + 39\cdot 47^{7} + 41\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 39 + 17\cdot 47 + 37\cdot 47^{2} + 40\cdot 47^{3} + 32\cdot 47^{4} + 41\cdot 47^{5} + 31\cdot 47^{6} + 9\cdot 47^{7} + 4\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 46 + 26\cdot 47 + 11\cdot 47^{2} + 5\cdot 47^{3} + 37\cdot 47^{4} + 4\cdot 47^{5} + 21\cdot 47^{6} + 33\cdot 47^{7} + 14\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display

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

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

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$$4$$(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.