Properties

Label 2.92416.8t5.b.a
Dimension $2$
Group $Q_8$
Conductor $92416$
Root number $-1$
Indicator $-1$

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Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(92416\)\(\medspace = 2^{8} \cdot 19^{2}\)
Frobenius-Schur indicator: $-1$
Root number: $-1$
Artin number field: Galois closure of 8.0.789298907447296.1
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 \(\Q(\sqrt{2}, \sqrt{19})\)

Defining polynomial

$f(x)$$=$$ x^{8} + 76 x^{6} + 1748 x^{4} + 12996 x^{2} + 29241 $.

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 11.

Roots:
$r_{ 1 }$ $=$ $ 22 + 12\cdot 73 + 32\cdot 73^{2} + 22\cdot 73^{3} + 35\cdot 73^{4} + 25\cdot 73^{5} + 51\cdot 73^{6} + 62\cdot 73^{7} + 34\cdot 73^{8} + 63\cdot 73^{9} + 27\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 2 }$ $=$ $ 23 + 15\cdot 73 + 65\cdot 73^{2} + 9\cdot 73^{3} + 40\cdot 73^{4} + 56\cdot 73^{5} + 11\cdot 73^{6} + 2\cdot 73^{7} + 7\cdot 73^{8} + 3\cdot 73^{9} + 40\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 3 }$ $=$ $ 25 + 34\cdot 73 + 72\cdot 73^{2} + 40\cdot 73^{3} + 42\cdot 73^{4} + 40\cdot 73^{5} + 39\cdot 73^{6} + 37\cdot 73^{7} + 46\cdot 73^{8} + 8\cdot 73^{9} + 45\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 4 }$ $=$ $ 29 + 46\cdot 73 + 17\cdot 73^{2} + 7\cdot 73^{3} + 41\cdot 73^{4} + 50\cdot 73^{5} + 34\cdot 73^{6} + 46\cdot 73^{7} + 70\cdot 73^{8} + 52\cdot 73^{9} + 48\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 5 }$ $=$ $ 44 + 26\cdot 73 + 55\cdot 73^{2} + 65\cdot 73^{3} + 31\cdot 73^{4} + 22\cdot 73^{5} + 38\cdot 73^{6} + 26\cdot 73^{7} + 2\cdot 73^{8} + 20\cdot 73^{9} + 24\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 6 }$ $=$ $ 48 + 38\cdot 73 + 32\cdot 73^{3} + 30\cdot 73^{4} + 32\cdot 73^{5} + 33\cdot 73^{6} + 35\cdot 73^{7} + 26\cdot 73^{8} + 64\cdot 73^{9} + 27\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 7 }$ $=$ $ 50 + 57\cdot 73 + 7\cdot 73^{2} + 63\cdot 73^{3} + 32\cdot 73^{4} + 16\cdot 73^{5} + 61\cdot 73^{6} + 70\cdot 73^{7} + 65\cdot 73^{8} + 69\cdot 73^{9} + 32\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 8 }$ $=$ $ 51 + 60\cdot 73 + 40\cdot 73^{2} + 50\cdot 73^{3} + 37\cdot 73^{4} + 47\cdot 73^{5} + 21\cdot 73^{6} + 10\cdot 73^{7} + 38\cdot 73^{8} + 9\cdot 73^{9} + 45\cdot 73^{10} +O\left(73^{ 11 }\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$$4$$(1,6,8,3)(2,5,7,4)$$0$
$2$$4$$(1,2,8,7)(3,5,6,4)$$0$
$2$$4$$(1,4,8,5)(2,6,7,3)$$0$

The blue line marks the conjugacy class containing complex conjugation.