Properties

Label 2.57600.8t5.e.a
Dimension $2$
Group $Q_8$
Conductor $57600$
Root number $1$
Indicator $-1$

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Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(57600\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2}\)
Frobenius-Schur indicator: $-1$
Root number: $1$
Artin number field: Galois closure of 8.8.7644119040000.1
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} - 60 x^{6} + 900 x^{4} - 4500 x^{2} + 5625 $.

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 12.

Roots:
$r_{ 1 }$ $=$ $ 3 + 43\cdot 47 + 29\cdot 47^{2} + 8\cdot 47^{3} + 3\cdot 47^{4} + 8\cdot 47^{5} + 44\cdot 47^{6} + 17\cdot 47^{7} + 5\cdot 47^{9} + 26\cdot 47^{10} + 18\cdot 47^{11} +O\left(47^{ 12 }\right)$
$r_{ 2 }$ $=$ $ 18 + 6\cdot 47 + 34\cdot 47^{2} + 17\cdot 47^{3} + 32\cdot 47^{4} + 43\cdot 47^{5} + 23\cdot 47^{6} + 33\cdot 47^{7} + 31\cdot 47^{8} + 17\cdot 47^{9} + 22\cdot 47^{10} + 26\cdot 47^{11} +O\left(47^{ 12 }\right)$
$r_{ 3 }$ $=$ $ 19 + 23\cdot 47 + 21\cdot 47^{2} + 33\cdot 47^{3} + 16\cdot 47^{5} + 36\cdot 47^{6} + 36\cdot 47^{7} + 32\cdot 47^{8} + 13\cdot 47^{9} + 43\cdot 47^{10} + 34\cdot 47^{11} +O\left(47^{ 12 }\right)$
$r_{ 4 }$ $=$ $ 20 + 47 + 5\cdot 47^{2} + 23\cdot 47^{3} + 11\cdot 47^{4} + 6\cdot 47^{5} + 14\cdot 47^{6} + 18\cdot 47^{7} + 23\cdot 47^{8} + 7\cdot 47^{9} + 22\cdot 47^{10} + 2\cdot 47^{11} +O\left(47^{ 12 }\right)$
$r_{ 5 }$ $=$ $ 27 + 45\cdot 47 + 41\cdot 47^{2} + 23\cdot 47^{3} + 35\cdot 47^{4} + 40\cdot 47^{5} + 32\cdot 47^{6} + 28\cdot 47^{7} + 23\cdot 47^{8} + 39\cdot 47^{9} + 24\cdot 47^{10} + 44\cdot 47^{11} +O\left(47^{ 12 }\right)$
$r_{ 6 }$ $=$ $ 28 + 23\cdot 47 + 25\cdot 47^{2} + 13\cdot 47^{3} + 46\cdot 47^{4} + 30\cdot 47^{5} + 10\cdot 47^{6} + 10\cdot 47^{7} + 14\cdot 47^{8} + 33\cdot 47^{9} + 3\cdot 47^{10} + 12\cdot 47^{11} +O\left(47^{ 12 }\right)$
$r_{ 7 }$ $=$ $ 29 + 40\cdot 47 + 12\cdot 47^{2} + 29\cdot 47^{3} + 14\cdot 47^{4} + 3\cdot 47^{5} + 23\cdot 47^{6} + 13\cdot 47^{7} + 15\cdot 47^{8} + 29\cdot 47^{9} + 24\cdot 47^{10} + 20\cdot 47^{11} +O\left(47^{ 12 }\right)$
$r_{ 8 }$ $=$ $ 44 + 3\cdot 47 + 17\cdot 47^{2} + 38\cdot 47^{3} + 43\cdot 47^{4} + 38\cdot 47^{5} + 2\cdot 47^{6} + 29\cdot 47^{7} + 46\cdot 47^{8} + 41\cdot 47^{9} + 20\cdot 47^{10} + 28\cdot 47^{11} +O\left(47^{ 12 }\right)$

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

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

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

The blue line marks the conjugacy class containing complex conjugation.