Properties

Label 2.176400.8t5.b.a
Dimension $2$
Group $Q_8$
Conductor $176400$
Root number $1$
Indicator $-1$

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Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(176400\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \)
Frobenius-Schur indicator: $-1$
Root number: $1$
Artin field: Galois closure of 8.0.343064484000000.1
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 \(\Q(\sqrt{5}, \sqrt{21})\)

Defining polynomial

$f(x)$$=$ \( x^{8} + 105x^{6} + 3780x^{4} + 55125x^{2} + 275625 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 10.

Roots:
$r_{ 1 }$ $=$ \( 5 + 21\cdot 41 + 10\cdot 41^{2} + 24\cdot 41^{3} + 13\cdot 41^{4} + 27\cdot 41^{5} + 19\cdot 41^{6} + 10\cdot 41^{7} + 8\cdot 41^{8} + 12\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 29\cdot 41 + 22\cdot 41^{2} + 8\cdot 41^{3} + 9\cdot 41^{4} + 5\cdot 41^{5} + 6\cdot 41^{6} + 32\cdot 41^{7} + 33\cdot 41^{8} + 22\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 10 + 31\cdot 41 + 10\cdot 41^{2} + 31\cdot 41^{3} + 16\cdot 41^{4} + 38\cdot 41^{5} + 19\cdot 41^{6} + 5\cdot 41^{7} + 4\cdot 41^{8} + 10\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 12 + 12\cdot 41 + 31\cdot 41^{2} + 39\cdot 41^{3} + 37\cdot 41^{4} + 25\cdot 41^{5} + 11\cdot 41^{6} + 16\cdot 41^{7} + 14\cdot 41^{8} + 23\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 29 + 28\cdot 41 + 9\cdot 41^{2} + 41^{3} + 3\cdot 41^{4} + 15\cdot 41^{5} + 29\cdot 41^{6} + 24\cdot 41^{7} + 26\cdot 41^{8} + 17\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 31 + 9\cdot 41 + 30\cdot 41^{2} + 9\cdot 41^{3} + 24\cdot 41^{4} + 2\cdot 41^{5} + 21\cdot 41^{6} + 35\cdot 41^{7} + 36\cdot 41^{8} + 30\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 35 + 11\cdot 41 + 18\cdot 41^{2} + 32\cdot 41^{3} + 31\cdot 41^{4} + 35\cdot 41^{5} + 34\cdot 41^{6} + 8\cdot 41^{7} + 7\cdot 41^{8} + 18\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 36 + 19\cdot 41 + 30\cdot 41^{2} + 16\cdot 41^{3} + 27\cdot 41^{4} + 13\cdot 41^{5} + 21\cdot 41^{6} + 30\cdot 41^{7} + 32\cdot 41^{8} + 28\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character valueComplex conjugation
$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$