Properties

Label 2.2e8_3e2_5e2.8t5.6c1
Dimension 2
Group $Q_8$
Conductor $ 2^{8} \cdot 3^{2} \cdot 5^{2}$
Root number 1
Frobenius-Schur indicator -1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$57600= 2^{8} \cdot 3^{2} \cdot 5^{2} $
Artin number field: Splitting field of $f= x^{8} + 60 x^{6} + 1170 x^{4} + 9000 x^{2} + 22500 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $Q_8$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

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.