Properties

Label 2.1575.8t11.a
Dimension 2
Group $Q_8:C_2$
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 $
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:$1575= 3^{2} \cdot 5^{2} \cdot 7 $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 10 x^{6} + 11 x^{5} + 34 x^{4} - 29 x^{3} - 55 x^{2} + 4 x + 76 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $Q_8:C_2$
Parity: Odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 22 + 82\cdot 109 + 67\cdot 109^{2} + 26\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 28 + 48\cdot 109 + 52\cdot 109^{2} + 88\cdot 109^{3} + 101\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 37 + 97\cdot 109 + 26\cdot 109^{2} + 84\cdot 109^{3} + 23\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 40 + 108\cdot 109 + 14\cdot 109^{2} + 80\cdot 109^{3} + 55\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 48 + 54\cdot 109 + 82\cdot 109^{2} + 7\cdot 109^{3} + 101\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 83 + 92\cdot 109 + 99\cdot 109^{2} + 3\cdot 109^{3} + 65\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 89 + 6\cdot 109 + 6\cdot 109^{2} + 68\cdot 109^{3} + 35\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 90 + 54\cdot 109 + 85\cdot 109^{2} + 102\cdot 109^{3} + 26\cdot 109^{4} +O\left(109^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$1$ $2$ $(1,2)(3,5)(4,6)(7,8)$ $-2$ $-2$
$2$ $2$ $(1,3)(2,5)(4,7)(6,8)$ $0$ $0$
$2$ $2$ $(3,5)(7,8)$ $0$ $0$
$2$ $2$ $(1,7)(2,8)(3,6)(4,5)$ $0$ $0$
$1$ $4$ $(1,4,2,6)(3,7,5,8)$ $-2 \zeta_{4}$ $2 \zeta_{4}$
$1$ $4$ $(1,6,2,4)(3,8,5,7)$ $2 \zeta_{4}$ $-2 \zeta_{4}$
$2$ $4$ $(1,7,2,8)(3,4,5,6)$ $0$ $0$
$2$ $4$ $(1,5,2,3)(4,8,6,7)$ $0$ $0$
$2$ $4$ $(1,4,2,6)(3,8,5,7)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.