Properties

Label 2.3e2_7e2_17e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 3^{2} \cdot 7^{2} \cdot 17^{2}$
Root number -1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$127449= 3^{2} \cdot 7^{2} \cdot 17^{2} $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} - 110 x^{6} + 153 x^{5} + 3789 x^{4} + 1989 x^{3} - 44000 x^{2} - 97899 x - 46703 $ 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 5.
Roots:
$r_{ 1 }$ $=$ $ 1 + 44\cdot 47 + 27\cdot 47^{2} + 36\cdot 47^{3} + 21\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 5 + 6\cdot 47^{3} + 36\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 18 + 25\cdot 47 + 25\cdot 47^{2} + 14\cdot 47^{3} + 44\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 22 + 2\cdot 47 + 30\cdot 47^{2} + 27\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 27 + 43\cdot 47 + 34\cdot 47^{2} + 11\cdot 47^{3} + 30\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 35 + 3\cdot 47 + 38\cdot 47^{2} + 10\cdot 47^{3} + 16\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 39 + 42\cdot 47 + 25\cdot 47^{2} + 7\cdot 47^{3} + 8\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 44 + 25\cdot 47 + 5\cdot 47^{2} + 6\cdot 47^{3} + 4\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,3)(4,6)(5,7)$$-2$
$2$$4$$(1,3,8,2)(4,5,6,7)$$0$
$2$$4$$(1,7,8,5)(2,4,3,6)$$0$
$2$$4$$(1,6,8,4)(2,7,3,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.