Properties

Label 2.120.8t11.c
Dimension $2$
Group $Q_8:C_2$
Conductor $120$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Artin number field: Galois closure of 8.0.3240000.1
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-6}, \sqrt{10})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 7 + 19\cdot 79 + 44\cdot 79^{2} + 40\cdot 79^{3} + 47\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 17 + 74\cdot 79 + 66\cdot 79^{2} + 61\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 19 + 78\cdot 79 + 12\cdot 79^{2} + 15\cdot 79^{3} + 56\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 32 + 66\cdot 79 + 21\cdot 79^{2} + 70\cdot 79^{3} + 68\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 33 + 44\cdot 79 + 78\cdot 79^{2} + 65\cdot 79^{3} + 37\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 38 + 72\cdot 79 + 68\cdot 79^{2} + 13\cdot 79^{3} + 12\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 40 + 45\cdot 79 + 63\cdot 79^{2} + 67\cdot 79^{3} + 17\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 54 + 73\cdot 79 + 37\cdot 79^{2} + 59\cdot 79^{3} + 11\cdot 79^{4} +O(79^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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