Properties

Label 2.2e8_3e2_11e2.8t5.7c1
Dimension 2
Group $Q_8$
Conductor $ 2^{8} \cdot 3^{2} \cdot 11^{2}$
Root number -1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$278784= 2^{8} \cdot 3^{2} \cdot 11^{2} $
Artin number field: Splitting field of $f= x^{8} - 132 x^{6} + 4356 x^{4} - 47916 x^{2} + 131769 $ 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 $\Q_{ 47 }$ to precision 13.
Roots:
$r_{ 1 }$ $=$ $ 1 + 2\cdot 47 + 25\cdot 47^{2} + 6\cdot 47^{3} + 3\cdot 47^{4} + 40\cdot 47^{5} + 46\cdot 47^{6} + 43\cdot 47^{7} + 39\cdot 47^{8} + 44\cdot 47^{9} + 18\cdot 47^{10} + 25\cdot 47^{11} + 16\cdot 47^{12} +O\left(47^{ 13 }\right)$
$r_{ 2 }$ $=$ $ 6 + 22\cdot 47 + 25\cdot 47^{2} + 8\cdot 47^{3} + 45\cdot 47^{4} + 6\cdot 47^{5} + 37\cdot 47^{6} + 19\cdot 47^{7} + 5\cdot 47^{8} + 3\cdot 47^{9} + 46\cdot 47^{11} + 46\cdot 47^{12} +O\left(47^{ 13 }\right)$
$r_{ 3 }$ $=$ $ 9 + 45\cdot 47 + 35\cdot 47^{2} + 41\cdot 47^{3} + 30\cdot 47^{4} + 27\cdot 47^{5} + 21\cdot 47^{6} + 5\cdot 47^{7} + 47^{8} + 29\cdot 47^{9} + 40\cdot 47^{10} + 23\cdot 47^{11} + 47^{12} +O\left(47^{ 13 }\right)$
$r_{ 4 }$ $=$ $ 22 + 18\cdot 47 + 47^{2} + 17\cdot 47^{3} + 47^{4} + 26\cdot 47^{5} + 9\cdot 47^{6} + 32\cdot 47^{7} + 9\cdot 47^{8} + 17\cdot 47^{9} + 17\cdot 47^{10} + 33\cdot 47^{11} + 45\cdot 47^{12} +O\left(47^{ 13 }\right)$
$r_{ 5 }$ $=$ $ 25 + 28\cdot 47 + 45\cdot 47^{2} + 29\cdot 47^{3} + 45\cdot 47^{4} + 20\cdot 47^{5} + 37\cdot 47^{6} + 14\cdot 47^{7} + 37\cdot 47^{8} + 29\cdot 47^{9} + 29\cdot 47^{10} + 13\cdot 47^{11} + 47^{12} +O\left(47^{ 13 }\right)$
$r_{ 6 }$ $=$ $ 38 + 47 + 11\cdot 47^{2} + 5\cdot 47^{3} + 16\cdot 47^{4} + 19\cdot 47^{5} + 25\cdot 47^{6} + 41\cdot 47^{7} + 45\cdot 47^{8} + 17\cdot 47^{9} + 6\cdot 47^{10} + 23\cdot 47^{11} + 45\cdot 47^{12} +O\left(47^{ 13 }\right)$
$r_{ 7 }$ $=$ $ 41 + 24\cdot 47 + 21\cdot 47^{2} + 38\cdot 47^{3} + 47^{4} + 40\cdot 47^{5} + 9\cdot 47^{6} + 27\cdot 47^{7} + 41\cdot 47^{8} + 43\cdot 47^{9} + 46\cdot 47^{10} +O\left(47^{ 13 }\right)$
$r_{ 8 }$ $=$ $ 46 + 44\cdot 47 + 21\cdot 47^{2} + 40\cdot 47^{3} + 43\cdot 47^{4} + 6\cdot 47^{5} + 3\cdot 47^{7} + 7\cdot 47^{8} + 2\cdot 47^{9} + 28\cdot 47^{10} + 21\cdot 47^{11} + 30\cdot 47^{12} +O\left(47^{ 13 }\right)$

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

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

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$$(1,4,8,5)(2,6,7,3)$$0$
$2$$4$$(1,6,8,3)(2,5,7,4)$$0$
The blue line marks the conjugacy class containing complex conjugation.