Properties

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

Related objects

Learn more about

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 8.8.47775744000000.3
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 $\Q_{ 19 }$ to precision 14.

Roots:
$r_{ 1 }$ $=$ $ 1 + 6\cdot 19 + 14\cdot 19^{2} + 19^{3} + 12\cdot 19^{4} + 13\cdot 19^{5} + 18\cdot 19^{6} + 9\cdot 19^{8} + 8\cdot 19^{9} + 12\cdot 19^{10} + 5\cdot 19^{11} + 8\cdot 19^{13} +O\left(19^{ 14 }\right)$
$r_{ 2 }$ $=$ $ 4 + 7\cdot 19 + 3\cdot 19^{2} + 15\cdot 19^{3} + 19^{4} + 10\cdot 19^{5} + 13\cdot 19^{6} + 17\cdot 19^{7} + 10\cdot 19^{8} + 11\cdot 19^{9} + 10\cdot 19^{10} + 10\cdot 19^{11} + 3\cdot 19^{12} + 12\cdot 19^{13} +O\left(19^{ 14 }\right)$
$r_{ 3 }$ $=$ $ 6 + 12\cdot 19 + 19^{2} + 18\cdot 19^{3} + 13\cdot 19^{4} + 13\cdot 19^{5} + 4\cdot 19^{6} + 10\cdot 19^{7} + 2\cdot 19^{9} + 15\cdot 19^{10} + 11\cdot 19^{11} + 17\cdot 19^{12} + 14\cdot 19^{13} +O\left(19^{ 14 }\right)$
$r_{ 4 }$ $=$ $ 8 + 2\cdot 19 + 2\cdot 19^{2} + 3\cdot 19^{3} + 5\cdot 19^{4} + 10\cdot 19^{5} + 9\cdot 19^{7} + 17\cdot 19^{8} + 12\cdot 19^{9} + 9\cdot 19^{11} + 3\cdot 19^{12} + 3\cdot 19^{13} +O\left(19^{ 14 }\right)$
$r_{ 5 }$ $=$ $ 11 + 16\cdot 19 + 16\cdot 19^{2} + 15\cdot 19^{3} + 13\cdot 19^{4} + 8\cdot 19^{5} + 18\cdot 19^{6} + 9\cdot 19^{7} + 19^{8} + 6\cdot 19^{9} + 18\cdot 19^{10} + 9\cdot 19^{11} + 15\cdot 19^{12} + 15\cdot 19^{13} +O\left(19^{ 14 }\right)$
$r_{ 6 }$ $=$ $ 13 + 6\cdot 19 + 17\cdot 19^{2} + 5\cdot 19^{4} + 5\cdot 19^{5} + 14\cdot 19^{6} + 8\cdot 19^{7} + 18\cdot 19^{8} + 16\cdot 19^{9} + 3\cdot 19^{10} + 7\cdot 19^{11} + 19^{12} + 4\cdot 19^{13} +O\left(19^{ 14 }\right)$
$r_{ 7 }$ $=$ $ 15 + 11\cdot 19 + 15\cdot 19^{2} + 3\cdot 19^{3} + 17\cdot 19^{4} + 8\cdot 19^{5} + 5\cdot 19^{6} + 19^{7} + 8\cdot 19^{8} + 7\cdot 19^{9} + 8\cdot 19^{10} + 8\cdot 19^{11} + 15\cdot 19^{12} + 6\cdot 19^{13} +O\left(19^{ 14 }\right)$
$r_{ 8 }$ $=$ $ 18 + 12\cdot 19 + 4\cdot 19^{2} + 17\cdot 19^{3} + 6\cdot 19^{4} + 5\cdot 19^{5} + 18\cdot 19^{7} + 9\cdot 19^{8} + 10\cdot 19^{9} + 6\cdot 19^{10} + 13\cdot 19^{11} + 18\cdot 19^{12} + 10\cdot 19^{13} +O\left(19^{ 14 }\right)$

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

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

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

The blue line marks the conjugacy class containing complex conjugation.