Properties

Label 2.92416.8t5.a.a
Dimension $2$
Group $Q_8$
Conductor $92416$
Root number $1$
Indicator $-1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(92416\)\(\medspace = 2^{8} \cdot 19^{2} \)
Frobenius-Schur indicator: $-1$
Root number: $1$
Artin field: Galois closure of 8.8.789298907447296.1
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 \(\Q(\sqrt{2}, \sqrt{19})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 76x^{6} + 1748x^{4} - 12996x^{2} + 29241 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 10.

Roots:
$r_{ 1 }$ $=$ \( 1 + 30\cdot 31 + 2\cdot 31^{2} + 24\cdot 31^{3} + 3\cdot 31^{4} + 13\cdot 31^{5} + 2\cdot 31^{6} + 10\cdot 31^{7} + 7\cdot 31^{8} + 18\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 + 2\cdot 31 + 20\cdot 31^{2} + 15\cdot 31^{3} + 24\cdot 31^{4} + 17\cdot 31^{5} + 26\cdot 31^{6} + 19\cdot 31^{7} + 26\cdot 31^{8} + 9\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 4 + 24\cdot 31 + 4\cdot 31^{2} + 18\cdot 31^{3} + 26\cdot 31^{4} + 22\cdot 31^{5} + 13\cdot 31^{6} + 9\cdot 31^{7} + 6\cdot 31^{8} + 4\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 9 + 18\cdot 31 + 2\cdot 31^{2} + 29\cdot 31^{3} + 18\cdot 31^{4} + 13\cdot 31^{5} + 21\cdot 31^{6} + 16\cdot 31^{7} + 29\cdot 31^{8} + 7\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 + 12\cdot 31 + 28\cdot 31^{2} + 31^{3} + 12\cdot 31^{4} + 17\cdot 31^{5} + 9\cdot 31^{6} + 14\cdot 31^{7} + 31^{8} + 23\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 27 + 6\cdot 31 + 26\cdot 31^{2} + 12\cdot 31^{3} + 4\cdot 31^{4} + 8\cdot 31^{5} + 17\cdot 31^{6} + 21\cdot 31^{7} + 24\cdot 31^{8} + 26\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 28 + 28\cdot 31 + 10\cdot 31^{2} + 15\cdot 31^{3} + 6\cdot 31^{4} + 13\cdot 31^{5} + 4\cdot 31^{6} + 11\cdot 31^{7} + 4\cdot 31^{8} + 21\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 30 + 28\cdot 31^{2} + 6\cdot 31^{3} + 27\cdot 31^{4} + 17\cdot 31^{5} + 28\cdot 31^{6} + 20\cdot 31^{7} + 23\cdot 31^{8} + 12\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display

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

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

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$$4$$(1,6,8,3)(2,4,7,5)$$0$
$2$$4$$(1,5,8,4)(2,6,7,3)$$0$
$2$$4$$(1,7,8,2)(3,5,6,4)$$0$

The blue line marks the conjugacy class containing complex conjugation.