Properties

Label 2.2e8_3e2_7e2.8t5.9c1
Dimension 2
Group $Q_8$
Conductor $ 2^{8} \cdot 3^{2} \cdot 7^{2}$
Root number 1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$112896= 2^{8} \cdot 3^{2} \cdot 7^{2} $
Artin number field: Splitting field of $f= x^{8} - 84 x^{6} + 2394 x^{4} - 24696 x^{2} + 44100 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $Q_8$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 11.
Roots:
$r_{ 1 }$ $=$ $ 1 + 22\cdot 43 + 9\cdot 43^{2} + 13\cdot 43^{3} + 7\cdot 43^{4} + 9\cdot 43^{5} + 33\cdot 43^{6} + 35\cdot 43^{7} + 24\cdot 43^{8} + 30\cdot 43^{9} + 7\cdot 43^{10} +O\left(43^{ 11 }\right)$
$r_{ 2 }$ $=$ $ 2 + 27\cdot 43 + 2\cdot 43^{2} + 42\cdot 43^{3} + 32\cdot 43^{4} + 26\cdot 43^{5} + 15\cdot 43^{6} + 4\cdot 43^{7} + 15\cdot 43^{8} + 6\cdot 43^{9} + 5\cdot 43^{10} +O\left(43^{ 11 }\right)$
$r_{ 3 }$ $=$ $ 5 + 37\cdot 43 + 10\cdot 43^{2} + 2\cdot 43^{3} + 41\cdot 43^{4} + 30\cdot 43^{5} + 27\cdot 43^{6} + 24\cdot 43^{7} + 2\cdot 43^{8} + 3\cdot 43^{9} + 3\cdot 43^{10} +O\left(43^{ 11 }\right)$
$r_{ 4 }$ $=$ $ 21 + 15\cdot 43 + 36\cdot 43^{2} + 19\cdot 43^{3} + 38\cdot 43^{4} + 11\cdot 43^{5} + 14\cdot 43^{6} + 18\cdot 43^{7} + 36\cdot 43^{8} + 28\cdot 43^{9} + 41\cdot 43^{10} +O\left(43^{ 11 }\right)$
$r_{ 5 }$ $=$ $ 22 + 27\cdot 43 + 6\cdot 43^{2} + 23\cdot 43^{3} + 4\cdot 43^{4} + 31\cdot 43^{5} + 28\cdot 43^{6} + 24\cdot 43^{7} + 6\cdot 43^{8} + 14\cdot 43^{9} + 43^{10} +O\left(43^{ 11 }\right)$
$r_{ 6 }$ $=$ $ 38 + 5\cdot 43 + 32\cdot 43^{2} + 40\cdot 43^{3} + 43^{4} + 12\cdot 43^{5} + 15\cdot 43^{6} + 18\cdot 43^{7} + 40\cdot 43^{8} + 39\cdot 43^{9} + 39\cdot 43^{10} +O\left(43^{ 11 }\right)$
$r_{ 7 }$ $=$ $ 41 + 15\cdot 43 + 40\cdot 43^{2} + 10\cdot 43^{4} + 16\cdot 43^{5} + 27\cdot 43^{6} + 38\cdot 43^{7} + 27\cdot 43^{8} + 36\cdot 43^{9} + 37\cdot 43^{10} +O\left(43^{ 11 }\right)$
$r_{ 8 }$ $=$ $ 42 + 20\cdot 43 + 33\cdot 43^{2} + 29\cdot 43^{3} + 35\cdot 43^{4} + 33\cdot 43^{5} + 9\cdot 43^{6} + 7\cdot 43^{7} + 18\cdot 43^{8} + 12\cdot 43^{9} + 35\cdot 43^{10} +O\left(43^{ 11 }\right)$

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.