Properties

Label 2.112896.8t5.e
Dimension $2$
Group $Q_8$
Conductor $112896$
Indicator $-1$

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:\(112896\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7^{2} \)
Frobenius-Schur indicator: $-1$
Root number: $-1$
Artin number field: Galois closure of 8.8.29365647704064.1
Galois orbit size: $1$
Smallest permutation container: $Q_8$
Parity: even
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{3})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 10.
Roots:
$r_{ 1 }$ $=$ \( 3 + 66\cdot 73 + 3\cdot 73^{2} + 32\cdot 73^{3} + 60\cdot 73^{4} + 65\cdot 73^{5} + 29\cdot 73^{6} + 10\cdot 73^{7} + 24\cdot 73^{8} + 19\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 11 + 48\cdot 73 + 43\cdot 73^{2} + 45\cdot 73^{3} + 46\cdot 73^{4} + 15\cdot 73^{5} + 51\cdot 73^{6} + 50\cdot 73^{7} + 19\cdot 73^{8} + 30\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 20 + 32\cdot 73 + 36\cdot 73^{2} + 44\cdot 73^{3} + 2\cdot 73^{4} + 26\cdot 73^{5} + 52\cdot 73^{6} + 35\cdot 73^{7} + 53\cdot 73^{8} + 71\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 24 + 31\cdot 73 + 53\cdot 73^{2} + 45\cdot 73^{3} + 33\cdot 73^{4} + 34\cdot 73^{5} + 61\cdot 73^{6} + 44\cdot 73^{7} + 73^{8} + 62\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 49 + 41\cdot 73 + 19\cdot 73^{2} + 27\cdot 73^{3} + 39\cdot 73^{4} + 38\cdot 73^{5} + 11\cdot 73^{6} + 28\cdot 73^{7} + 71\cdot 73^{8} + 10\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 53 + 40\cdot 73 + 36\cdot 73^{2} + 28\cdot 73^{3} + 70\cdot 73^{4} + 46\cdot 73^{5} + 20\cdot 73^{6} + 37\cdot 73^{7} + 19\cdot 73^{8} + 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 62 + 24\cdot 73 + 29\cdot 73^{2} + 27\cdot 73^{3} + 26\cdot 73^{4} + 57\cdot 73^{5} + 21\cdot 73^{6} + 22\cdot 73^{7} + 53\cdot 73^{8} + 42\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 70 + 6\cdot 73 + 69\cdot 73^{2} + 40\cdot 73^{3} + 12\cdot 73^{4} + 7\cdot 73^{5} + 43\cdot 73^{6} + 62\cdot 73^{7} + 48\cdot 73^{8} + 53\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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