Properties

Label 2.73984.8t5.b.a
Dimension $2$
Group $Q_8$
Conductor $73984$
Root number $-1$
Indicator $-1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(73984\)\(\medspace = 2^{8} \cdot 17^{2} \)
Frobenius-Schur indicator: $-1$
Root number: $-1$
Artin field: Galois closure of 8.0.101240302206976.2
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 \(\Q(\sqrt{2}, \sqrt{17})\)

Defining polynomial

$f(x)$$=$ \( x^{8} + 68x^{6} + 986x^{4} + 4624x^{2} + 4624 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 10.

Roots:
$r_{ 1 }$ $=$ \( 10 + 11\cdot 47 + 8\cdot 47^{2} + 41\cdot 47^{3} + 15\cdot 47^{4} + 9\cdot 47^{5} + 19\cdot 47^{6} + 41\cdot 47^{7} + 6\cdot 47^{8} + 2\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 11 + 20\cdot 47 + 6\cdot 47^{2} + 19\cdot 47^{3} + 10\cdot 47^{4} + 45\cdot 47^{5} + 22\cdot 47^{6} + 25\cdot 47^{7} + 6\cdot 47^{8} + 2\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 14 + 45\cdot 47 + 2\cdot 47^{2} + 13\cdot 47^{3} + 44\cdot 47^{4} + 43\cdot 47^{5} + 12\cdot 47^{6} + 34\cdot 47^{7} + 26\cdot 47^{8} + 5\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 19 + 43\cdot 47 + 8\cdot 47^{2} + 9\cdot 47^{3} + 37\cdot 47^{4} + 6\cdot 47^{5} + 44\cdot 47^{6} + 2\cdot 47^{7} + 29\cdot 47^{8} + 21\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 28 + 3\cdot 47 + 38\cdot 47^{2} + 37\cdot 47^{3} + 9\cdot 47^{4} + 40\cdot 47^{5} + 2\cdot 47^{6} + 44\cdot 47^{7} + 17\cdot 47^{8} + 25\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 33 + 47 + 44\cdot 47^{2} + 33\cdot 47^{3} + 2\cdot 47^{4} + 3\cdot 47^{5} + 34\cdot 47^{6} + 12\cdot 47^{7} + 20\cdot 47^{8} + 41\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 36 + 26\cdot 47 + 40\cdot 47^{2} + 27\cdot 47^{3} + 36\cdot 47^{4} + 47^{5} + 24\cdot 47^{6} + 21\cdot 47^{7} + 40\cdot 47^{8} + 44\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 37 + 35\cdot 47 + 38\cdot 47^{2} + 5\cdot 47^{3} + 31\cdot 47^{4} + 37\cdot 47^{5} + 27\cdot 47^{6} + 5\cdot 47^{7} + 40\cdot 47^{8} + 44\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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