Properties

Label 2.119025.8t5.b
Dimension $2$
Group $Q_8$
Conductor $119025$
Indicator $-1$

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:\(119025\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 23^{2}\)
Frobenius-Schur indicator: $-1$
Root number: $-1$
Artin number field: Galois closure of 8.0.1686221298140625.1
Galois orbit size: $1$
Smallest permutation container: $Q_8$
Parity: even
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{69})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 5 + 7\cdot 139 + 19\cdot 139^{2} + 56\cdot 139^{3} + 131\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 + 73\cdot 139 + 25\cdot 139^{2} + 93\cdot 139^{3} + 78\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 30 + 53\cdot 139 + 93\cdot 139^{3} + 93\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 69 + 30\cdot 139 + 101\cdot 139^{2} + 94\cdot 139^{3} + 2\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 95 + 87\cdot 139 + 76\cdot 139^{2} + 116\cdot 139^{3} + 56\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 101 + 19\cdot 139 + 7\cdot 139^{2} + 85\cdot 139^{3} + 113\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 117 + 131\cdot 139 + 83\cdot 139^{2} + 53\cdot 139^{3} + 10\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 123 + 13\cdot 139 + 103\cdot 139^{2} + 102\cdot 139^{3} + 68\cdot 139^{4} +O\left(139^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$
$1$ $1$ $()$ $2$
$1$ $2$ $(1,4)(2,8)(3,5)(6,7)$ $-2$
$2$ $4$ $(1,2,4,8)(3,7,5,6)$ $0$
$2$ $4$ $(1,6,4,7)(2,5,8,3)$ $0$
$2$ $4$ $(1,5,4,3)(2,7,8,6)$ $0$
The blue line marks the conjugacy class containing complex conjugation.