Properties

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

Related objects

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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} + 810 x^{4} + 1800 x^{2} + 900 $ 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 $\Q_{ 71 }$ to precision 11.
Roots:
$r_{ 1 }$ $=$ $ 12 + 34\cdot 71 + 41\cdot 71^{2} + 46\cdot 71^{3} + 49\cdot 71^{4} + 46\cdot 71^{5} + 71^{6} + 4\cdot 71^{7} + 12\cdot 71^{8} + 46\cdot 71^{9} + 4\cdot 71^{10} +O\left(71^{ 11 }\right)$
$r_{ 2 }$ $=$ $ 21 + 11\cdot 71 + 49\cdot 71^{2} + 64\cdot 71^{3} + 60\cdot 71^{4} + 23\cdot 71^{5} + 9\cdot 71^{6} + 58\cdot 71^{7} + 45\cdot 71^{8} + 52\cdot 71^{9} + 45\cdot 71^{10} +O\left(71^{ 11 }\right)$
$r_{ 3 }$ $=$ $ 24 + 12\cdot 71 + 67\cdot 71^{2} + 24\cdot 71^{3} + 26\cdot 71^{4} + 25\cdot 71^{5} + 53\cdot 71^{6} + 60\cdot 71^{7} + 46\cdot 71^{8} + 48\cdot 71^{9} + 48\cdot 71^{10} +O\left(71^{ 11 }\right)$
$r_{ 4 }$ $=$ $ 25 + 5\cdot 71 + 64\cdot 71^{2} + 20\cdot 71^{3} + 28\cdot 71^{4} + 27\cdot 71^{5} + 35\cdot 71^{6} + 11\cdot 71^{7} + 32\cdot 71^{8} + 50\cdot 71^{9} + 63\cdot 71^{10} +O\left(71^{ 11 }\right)$
$r_{ 5 }$ $=$ $ 46 + 65\cdot 71 + 6\cdot 71^{2} + 50\cdot 71^{3} + 42\cdot 71^{4} + 43\cdot 71^{5} + 35\cdot 71^{6} + 59\cdot 71^{7} + 38\cdot 71^{8} + 20\cdot 71^{9} + 7\cdot 71^{10} +O\left(71^{ 11 }\right)$
$r_{ 6 }$ $=$ $ 47 + 58\cdot 71 + 3\cdot 71^{2} + 46\cdot 71^{3} + 44\cdot 71^{4} + 45\cdot 71^{5} + 17\cdot 71^{6} + 10\cdot 71^{7} + 24\cdot 71^{8} + 22\cdot 71^{9} + 22\cdot 71^{10} +O\left(71^{ 11 }\right)$
$r_{ 7 }$ $=$ $ 50 + 59\cdot 71 + 21\cdot 71^{2} + 6\cdot 71^{3} + 10\cdot 71^{4} + 47\cdot 71^{5} + 61\cdot 71^{6} + 12\cdot 71^{7} + 25\cdot 71^{8} + 18\cdot 71^{9} + 25\cdot 71^{10} +O\left(71^{ 11 }\right)$
$r_{ 8 }$ $=$ $ 59 + 36\cdot 71 + 29\cdot 71^{2} + 24\cdot 71^{3} + 21\cdot 71^{4} + 24\cdot 71^{5} + 69\cdot 71^{6} + 66\cdot 71^{7} + 58\cdot 71^{8} + 24\cdot 71^{9} + 66\cdot 71^{10} +O\left(71^{ 11 }\right)$

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

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

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.