Properties

Label 2.1560.8t11.f
Dimension $2$
Group $Q_8:C_2$
Conductor $1560$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:\(1560\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Artin number field: Galois closure of 8.0.3701505600.2
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-30}, \sqrt{-39})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 12 + 95\cdot 157 + 120\cdot 157^{2} + 75\cdot 157^{3} + 70\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 25 + 87\cdot 157 + 98\cdot 157^{2} + 13\cdot 157^{3} + 149\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 55 + 63\cdot 157 + 147\cdot 157^{2} + 84\cdot 157^{3} + 100\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 65 + 68\cdot 157 + 104\cdot 157^{2} + 139\cdot 157^{3} + 150\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 96 + 42\cdot 157 + 143\cdot 157^{2} + 4\cdot 157^{3} + 41\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 106 + 102\cdot 157 + 82\cdot 157^{2} + 126\cdot 157^{3} + 64\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 116 + 38\cdot 157 + 138\cdot 157^{2} + 16\cdot 157^{3} + 148\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 153 + 129\cdot 157 + 106\cdot 157^{2} + 8\cdot 157^{3} + 60\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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