Properties

Label 2.112896.8t5.f.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 8.0.29365647704064.1
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 \(\Q(\sqrt{2}, \sqrt{3})\)

Defining polynomial

$f(x)$$=$ \( x^{8} + 84x^{6} + 1764x^{4} + 12348x^{2} + 21609 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 10.

Roots:
$r_{ 1 }$ $=$ \( 2 + 10\cdot 23 + 5\cdot 23^{2} + 18\cdot 23^{5} + 10\cdot 23^{6} + 15\cdot 23^{7} + 4\cdot 23^{8} + 2\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 19\cdot 23 + 14\cdot 23^{2} + 22\cdot 23^{3} + 18\cdot 23^{4} + 2\cdot 23^{6} + 23^{7} + 23^{8} +O(23^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 10 + 18\cdot 23 + 22\cdot 23^{2} + 5\cdot 23^{3} + 14\cdot 23^{4} + 5\cdot 23^{5} + 4\cdot 23^{6} + 10\cdot 23^{7} + 11\cdot 23^{8} + 6\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 11 + 22\cdot 23 + 14\cdot 23^{2} + 18\cdot 23^{3} + 19\cdot 23^{4} + 19\cdot 23^{5} + 21\cdot 23^{6} + 19\cdot 23^{7} + 17\cdot 23^{8} + 14\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 12 + 8\cdot 23^{2} + 4\cdot 23^{3} + 3\cdot 23^{4} + 3\cdot 23^{5} + 23^{6} + 3\cdot 23^{7} + 5\cdot 23^{8} + 8\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 13 + 4\cdot 23 + 17\cdot 23^{3} + 8\cdot 23^{4} + 17\cdot 23^{5} + 18\cdot 23^{6} + 12\cdot 23^{7} + 11\cdot 23^{8} + 16\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 17 + 3\cdot 23 + 8\cdot 23^{2} + 4\cdot 23^{4} + 22\cdot 23^{5} + 20\cdot 23^{6} + 21\cdot 23^{7} + 21\cdot 23^{8} + 22\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 21 + 12\cdot 23 + 17\cdot 23^{2} + 22\cdot 23^{3} + 22\cdot 23^{4} + 4\cdot 23^{5} + 12\cdot 23^{6} + 7\cdot 23^{7} + 18\cdot 23^{8} + 20\cdot 23^{9} +O(23^{10})\) 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 value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-2$
$2$$4$$(1,2,8,7)(3,5,6,4)$$0$
$2$$4$$(1,4,8,5)(2,6,7,3)$$0$
$2$$4$$(1,6,8,3)(2,5,7,4)$$0$

The blue line marks the conjugacy class containing complex conjugation.