Properties

Label 2.57600.8t5.d.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 8.0.47775744000000.2
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 \(\Q(\sqrt{5}, \sqrt{6})\)

Defining polynomial

$f(x)$$=$$ x^{8} + 60 x^{6} + 1170 x^{4} + 9000 x^{2} + 22500 $.

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 13.

Roots:
$r_{ 1 }$ $=$ $ 2 + 20\cdot 29 + 10\cdot 29^{2} + 10\cdot 29^{3} + 9\cdot 29^{4} + 7\cdot 29^{5} + 12\cdot 29^{6} + 15\cdot 29^{7} + 5\cdot 29^{8} + 21\cdot 29^{9} + 11\cdot 29^{10} + 7\cdot 29^{11} + 14\cdot 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 2 }$ $=$ $ 4 + 16\cdot 29 + 21\cdot 29^{2} + 18\cdot 29^{3} + 28\cdot 29^{4} + 18\cdot 29^{5} + 7\cdot 29^{6} + 27\cdot 29^{7} + 10\cdot 29^{8} + 19\cdot 29^{9} + 27\cdot 29^{10} + 8\cdot 29^{11} + 4\cdot 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 3 }$ $=$ $ 9 + 9\cdot 29 + 29^{2} + 2\cdot 29^{3} + 19\cdot 29^{5} + 29^{6} + 7\cdot 29^{7} + 8\cdot 29^{8} + 27\cdot 29^{9} + 6\cdot 29^{10} + 23\cdot 29^{11} + 21\cdot 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 4 }$ $=$ $ 10 + 26\cdot 29 + 18\cdot 29^{2} + 6\cdot 29^{3} + 27\cdot 29^{4} + 15\cdot 29^{5} + 14\cdot 29^{6} + 7\cdot 29^{7} + 25\cdot 29^{8} + 16\cdot 29^{9} + 27\cdot 29^{10} + 23\cdot 29^{11} + 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 5 }$ $=$ $ 19 + 2\cdot 29 + 10\cdot 29^{2} + 22\cdot 29^{3} + 29^{4} + 13\cdot 29^{5} + 14\cdot 29^{6} + 21\cdot 29^{7} + 3\cdot 29^{8} + 12\cdot 29^{9} + 29^{10} + 5\cdot 29^{11} + 27\cdot 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 6 }$ $=$ $ 20 + 19\cdot 29 + 27\cdot 29^{2} + 26\cdot 29^{3} + 28\cdot 29^{4} + 9\cdot 29^{5} + 27\cdot 29^{6} + 21\cdot 29^{7} + 20\cdot 29^{8} + 29^{9} + 22\cdot 29^{10} + 5\cdot 29^{11} + 7\cdot 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 7 }$ $=$ $ 25 + 12\cdot 29 + 7\cdot 29^{2} + 10\cdot 29^{3} + 10\cdot 29^{5} + 21\cdot 29^{6} + 29^{7} + 18\cdot 29^{8} + 9\cdot 29^{9} + 29^{10} + 20\cdot 29^{11} + 24\cdot 29^{12} +O\left(29^{ 13 }\right)$
$r_{ 8 }$ $=$ $ 27 + 8\cdot 29 + 18\cdot 29^{2} + 18\cdot 29^{3} + 19\cdot 29^{4} + 21\cdot 29^{5} + 16\cdot 29^{6} + 13\cdot 29^{7} + 23\cdot 29^{8} + 7\cdot 29^{9} + 17\cdot 29^{10} + 21\cdot 29^{11} + 14\cdot 29^{12} +O\left(29^{ 13 }\right)$

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

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

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

The blue line marks the conjugacy class containing complex conjugation.