Properties

Label 2.30976.8t5.b.a
Dimension $2$
Group $Q_8$
Conductor $30976$
Root number $-1$
Indicator $-1$

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Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(30976\)\(\medspace = 2^{8} \cdot 11^{2}\)
Frobenius-Schur indicator: $-1$
Root number: $-1$
Artin number field: Galois closure of 8.0.29721861554176.1
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 \(\Q(\sqrt{2}, \sqrt{11})\)

Defining polynomial

$f(x)$$=$$ x^{8} + 44 x^{6} + 308 x^{4} + 484 x^{2} + 121 $.

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 10.

Roots:
$r_{ 1 }$ $=$ $ 2 + 67\cdot 89 + 36\cdot 89^{2} + 87\cdot 89^{3} + 46\cdot 89^{4} + 33\cdot 89^{5} + 65\cdot 89^{6} + 57\cdot 89^{7} + 16\cdot 89^{8} + 46\cdot 89^{9} +O\left(89^{ 10 }\right)$
$r_{ 2 }$ $=$ $ 5 + 75\cdot 89 + 27\cdot 89^{2} + 73\cdot 89^{3} + 7\cdot 89^{4} + 11\cdot 89^{5} + 5\cdot 89^{6} + 3\cdot 89^{7} + 29\cdot 89^{8} + 22\cdot 89^{9} +O\left(89^{ 10 }\right)$
$r_{ 3 }$ $=$ $ 31 + 47\cdot 89 + 8\cdot 89^{2} + 54\cdot 89^{3} + 87\cdot 89^{4} + 88\cdot 89^{5} + 20\cdot 89^{6} + 3\cdot 89^{7} + 82\cdot 89^{8} + 69\cdot 89^{9} +O\left(89^{ 10 }\right)$
$r_{ 4 }$ $=$ $ 37 + 80\cdot 89 + 5\cdot 89^{2} + 44\cdot 89^{3} + 10\cdot 89^{4} + 46\cdot 89^{5} + 46\cdot 89^{6} + 81\cdot 89^{7} + 52\cdot 89^{8} + 18\cdot 89^{9} +O\left(89^{ 10 }\right)$
$r_{ 5 }$ $=$ $ 52 + 8\cdot 89 + 83\cdot 89^{2} + 44\cdot 89^{3} + 78\cdot 89^{4} + 42\cdot 89^{5} + 42\cdot 89^{6} + 7\cdot 89^{7} + 36\cdot 89^{8} + 70\cdot 89^{9} +O\left(89^{ 10 }\right)$
$r_{ 6 }$ $=$ $ 58 + 41\cdot 89 + 80\cdot 89^{2} + 34\cdot 89^{3} + 89^{4} + 68\cdot 89^{6} + 85\cdot 89^{7} + 6\cdot 89^{8} + 19\cdot 89^{9} +O\left(89^{ 10 }\right)$
$r_{ 7 }$ $=$ $ 84 + 13\cdot 89 + 61\cdot 89^{2} + 15\cdot 89^{3} + 81\cdot 89^{4} + 77\cdot 89^{5} + 83\cdot 89^{6} + 85\cdot 89^{7} + 59\cdot 89^{8} + 66\cdot 89^{9} +O\left(89^{ 10 }\right)$
$r_{ 8 }$ $=$ $ 87 + 21\cdot 89 + 52\cdot 89^{2} + 89^{3} + 42\cdot 89^{4} + 55\cdot 89^{5} + 23\cdot 89^{6} + 31\cdot 89^{7} + 72\cdot 89^{8} + 42\cdot 89^{9} +O\left(89^{ 10 }\right)$

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

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

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$$(1,6,8,3)(2,5,7,4)$$0$
$2$$4$$(1,5,8,4)(2,3,7,6)$$0$

The blue line marks the conjugacy class containing complex conjugation.