Properties

Label 2.5e2_61e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 5^{2} \cdot 61^{2}$
Root number -1
Frobenius-Schur indicator -1

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$93025= 5^{2} \cdot 61^{2} $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 57 x^{6} + 135 x^{5} + 306 x^{4} + 5365 x^{3} + 23383 x^{2} + 40951 x + 75421 $ 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 $\Q_{ 199 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 106\cdot 199 + 78\cdot 199^{2} + 61\cdot 199^{3} + 63\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 + 25\cdot 199 + 148\cdot 199^{2} + 111\cdot 199^{3} + 58\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 18 + 179\cdot 199 + 8\cdot 199^{2} + 14\cdot 199^{3} + 187\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 30 + 41\cdot 199 + 63\cdot 199^{2} + 50\cdot 199^{3} + 23\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 79 + 131\cdot 199 + 171\cdot 199^{2} + 154\cdot 199^{3} + 177\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 118 + 37\cdot 199 + 58\cdot 199^{2} + 80\cdot 199^{3} + 74\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 157 + 72\cdot 199 + 156\cdot 199^{2} + 13\cdot 199^{3} + 121\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 195 + 3\cdot 199 + 111\cdot 199^{2} + 110\cdot 199^{3} + 90\cdot 199^{4} +O\left(199^{ 5 }\right)$

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

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

Character values on conjugacy classes

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