Properties

Label 2.2e8_19e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 2^{8} \cdot 19^{2}$
Root number 1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$92416= 2^{8} \cdot 19^{2} $
Artin number field: Splitting field of $f= x^{8} - 76 x^{6} + 1748 x^{4} - 12996 x^{2} + 29241 $ 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 $\Q_{ 31 }$ to precision 13.
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} + 18\cdot 31^{10} + 26\cdot 31^{11} + 11\cdot 31^{12} +O\left(31^{ 13 }\right)$
$r_{ 2 }$ $=$ $ 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} + 31^{10} + 8\cdot 31^{11} + 29\cdot 31^{12} +O\left(31^{ 13 }\right)$
$r_{ 3 }$ $=$ $ 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} + 21\cdot 31^{10} + 24\cdot 31^{11} + 3\cdot 31^{12} +O\left(31^{ 13 }\right)$
$r_{ 4 }$ $=$ $ 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} + 25\cdot 31^{10} + 18\cdot 31^{11} + 13\cdot 31^{12} +O\left(31^{ 13 }\right)$
$r_{ 5 }$ $=$ $ 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} + 5\cdot 31^{10} + 12\cdot 31^{11} + 17\cdot 31^{12} +O\left(31^{ 13 }\right)$
$r_{ 6 }$ $=$ $ 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} + 9\cdot 31^{10} + 6\cdot 31^{11} + 27\cdot 31^{12} +O\left(31^{ 13 }\right)$
$r_{ 7 }$ $=$ $ 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} + 29\cdot 31^{10} + 22\cdot 31^{11} + 31^{12} +O\left(31^{ 13 }\right)$
$r_{ 8 }$ $=$ $ 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} + 12\cdot 31^{10} + 4\cdot 31^{11} + 19\cdot 31^{12} +O\left(31^{ 13 }\right)$

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

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

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.