Properties

Label 2.1280.8t11.a
Dimension $2$
Group $Q_8:C_2$
Conductor $1280$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:\(1280\)\(\medspace = 2^{8} \cdot 5 \)
Artin number field: Galois closure of 8.0.419430400.2
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(i, \sqrt{10})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ \( 2 + 23\cdot 41 + 3\cdot 41^{2} + 12\cdot 41^{3} + 38\cdot 41^{4} + 10\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 34\cdot 41 + 11\cdot 41^{2} + 37\cdot 41^{3} + 34\cdot 41^{4} + 13\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 14 + 41 + 26\cdot 41^{2} + 34\cdot 41^{3} + 18\cdot 41^{4} + 10\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 19 + 35\cdot 41 + 11\cdot 41^{2} + 8\cdot 41^{3} + 29\cdot 41^{4} + 20\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 + 5\cdot 41 + 29\cdot 41^{2} + 32\cdot 41^{3} + 11\cdot 41^{4} + 20\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 27 + 39\cdot 41 + 14\cdot 41^{2} + 6\cdot 41^{3} + 22\cdot 41^{4} + 30\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 36 + 6\cdot 41 + 29\cdot 41^{2} + 3\cdot 41^{3} + 6\cdot 41^{4} + 27\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 39 + 17\cdot 41 + 37\cdot 41^{2} + 28\cdot 41^{3} + 2\cdot 41^{4} + 30\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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