Properties

Label 2.112896.8t5.a.a
Dimension $2$
Group $Q_8$
Conductor $112896$
Root number $1$
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 field: Galois closure of 8.8.1438916737499136.1
Galois orbit size: $1$
Smallest permutation container: $Q_8$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{3}, \sqrt{14})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 84x^{6} + 2268x^{4} - 19404x^{2} + 441 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 10.

Roots:
$r_{ 1 }$ $=$ \( 4 + 30\cdot 47 + 2\cdot 47^{2} + 10\cdot 47^{3} + 24\cdot 47^{4} + 17\cdot 47^{5} + 34\cdot 47^{6} + 40\cdot 47^{7} + 41\cdot 47^{8} + 24\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 45\cdot 47 + 24\cdot 47^{2} + 25\cdot 47^{3} + 3\cdot 47^{5} + 6\cdot 47^{6} + 42\cdot 47^{7} + 45\cdot 47^{8} + 41\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 14 + 5\cdot 47 + 3\cdot 47^{2} + 6\cdot 47^{3} + 17\cdot 47^{4} + 45\cdot 47^{5} + 2\cdot 47^{6} + 32\cdot 47^{8} + 8\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 17 + 27\cdot 47 + 42\cdot 47^{2} + 25\cdot 47^{3} + 10\cdot 47^{4} + 40\cdot 47^{5} + 7\cdot 47^{6} + 37\cdot 47^{7} + 10\cdot 47^{8} + 27\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 30 + 19\cdot 47 + 4\cdot 47^{2} + 21\cdot 47^{3} + 36\cdot 47^{4} + 6\cdot 47^{5} + 39\cdot 47^{6} + 9\cdot 47^{7} + 36\cdot 47^{8} + 19\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 33 + 41\cdot 47 + 43\cdot 47^{2} + 40\cdot 47^{3} + 29\cdot 47^{4} + 47^{5} + 44\cdot 47^{6} + 46\cdot 47^{7} + 14\cdot 47^{8} + 38\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 37 + 47 + 22\cdot 47^{2} + 21\cdot 47^{3} + 46\cdot 47^{4} + 43\cdot 47^{5} + 40\cdot 47^{6} + 4\cdot 47^{7} + 47^{8} + 5\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 43 + 16\cdot 47 + 44\cdot 47^{2} + 36\cdot 47^{3} + 22\cdot 47^{4} + 29\cdot 47^{5} + 12\cdot 47^{6} + 6\cdot 47^{7} + 5\cdot 47^{8} + 22\cdot 47^{9} +O(47^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character valueComplex conjugation
$1$$1$$()$$2$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-2$
$2$$4$$(1,4,8,5)(2,6,7,3)$$0$
$2$$4$$(1,2,8,7)(3,5,6,4)$$0$
$2$$4$$(1,3,8,6)(2,4,7,5)$$0$