Properties

Label 2.112896.8t5.g.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 number field: Galois closure of 8.8.359729184374784.2
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 \(\Q(\sqrt{6}, \sqrt{14})\)

Defining polynomial

$f(x)$$=$$ x^{8} - 84 x^{6} + 1890 x^{4} - 10584 x^{2} + 1764 $.

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 11.

Roots:
$r_{ 1 }$ $=$ $ 13 + 62\cdot 67 + 27\cdot 67^{2} + 51\cdot 67^{3} + 5\cdot 67^{4} + 66\cdot 67^{5} + 36\cdot 67^{6} + 40\cdot 67^{7} + 3\cdot 67^{8} + 45\cdot 67^{9} + 57\cdot 67^{10} +O\left(67^{ 11 }\right)$
$r_{ 2 }$ $=$ $ 16 + 9\cdot 67 + 27\cdot 67^{2} + 2\cdot 67^{3} + 44\cdot 67^{4} + 14\cdot 67^{5} + 46\cdot 67^{6} + 25\cdot 67^{7} + 40\cdot 67^{8} + 63\cdot 67^{9} +O\left(67^{ 11 }\right)$
$r_{ 3 }$ $=$ $ 17 + 48\cdot 67 + 66\cdot 67^{2} + 53\cdot 67^{3} + 55\cdot 67^{4} + 16\cdot 67^{5} + 49\cdot 67^{6} + 40\cdot 67^{7} + 15\cdot 67^{8} + 26\cdot 67^{9} + 37\cdot 67^{10} +O\left(67^{ 11 }\right)$
$r_{ 4 }$ $=$ $ 24 + 7\cdot 67 + 65\cdot 67^{2} + 14\cdot 67^{3} + 63\cdot 67^{4} + 32\cdot 67^{5} + 44\cdot 67^{6} + 56\cdot 67^{7} + 54\cdot 67^{8} + 21\cdot 67^{9} + 58\cdot 67^{10} +O\left(67^{ 11 }\right)$
$r_{ 5 }$ $=$ $ 43 + 59\cdot 67 + 67^{2} + 52\cdot 67^{3} + 3\cdot 67^{4} + 34\cdot 67^{5} + 22\cdot 67^{6} + 10\cdot 67^{7} + 12\cdot 67^{8} + 45\cdot 67^{9} + 8\cdot 67^{10} +O\left(67^{ 11 }\right)$
$r_{ 6 }$ $=$ $ 50 + 18\cdot 67 + 13\cdot 67^{3} + 11\cdot 67^{4} + 50\cdot 67^{5} + 17\cdot 67^{6} + 26\cdot 67^{7} + 51\cdot 67^{8} + 40\cdot 67^{9} + 29\cdot 67^{10} +O\left(67^{ 11 }\right)$
$r_{ 7 }$ $=$ $ 51 + 57\cdot 67 + 39\cdot 67^{2} + 64\cdot 67^{3} + 22\cdot 67^{4} + 52\cdot 67^{5} + 20\cdot 67^{6} + 41\cdot 67^{7} + 26\cdot 67^{8} + 3\cdot 67^{9} + 66\cdot 67^{10} +O\left(67^{ 11 }\right)$
$r_{ 8 }$ $=$ $ 54 + 4\cdot 67 + 39\cdot 67^{2} + 15\cdot 67^{3} + 61\cdot 67^{4} + 30\cdot 67^{6} + 26\cdot 67^{7} + 63\cdot 67^{8} + 21\cdot 67^{9} + 9\cdot 67^{10} +O\left(67^{ 11 }\right)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.