Properties

Label 2.13e2_17e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 13^{2} \cdot 17^{2}$
Root number -1
Frobenius-Schur indicator -1

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$48841= 13^{2} \cdot 17^{2} $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} - 67 x^{6} - 16 x^{5} + 863 x^{4} + 1276 x^{3} + 392 x^{2} - 54 x + 1 $ 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 $\Q_{ 179 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 42 + 54\cdot 179 + 111\cdot 179^{2} + 6\cdot 179^{3} + 165\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 46 + 76\cdot 179 + 151\cdot 179^{2} + 6\cdot 179^{3} + 114\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 49 + 122\cdot 179 + 153\cdot 179^{2} + 19\cdot 179^{3} + 124\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 50 + 156\cdot 179 + 2\cdot 179^{2} + 169\cdot 179^{3} + 73\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 101 + 155\cdot 179 + 92\cdot 179^{2} + 101\cdot 179^{3} + 139\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 136 + 98\cdot 179 + 164\cdot 179^{2} + 124\cdot 179^{3} + 100\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 142 + 113\cdot 179 + 82\cdot 179^{2} + 47\cdot 179^{3} + 177\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 153 + 117\cdot 179 + 135\cdot 179^{2} + 60\cdot 179^{3} +O\left(179^{ 5 }\right)$

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

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

Character values on conjugacy classes

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