Properties

Label 2.2e8_3e2_7e2.8t5.7c1
Dimension 2
Group $Q_8$
Conductor $ 2^{8} \cdot 3^{2} \cdot 7^{2}$
Root number -1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$112896= 2^{8} \cdot 3^{2} \cdot 7^{2} $
Artin number field: Splitting field of $f= x^{8} - 84 x^{6} + 1764 x^{4} - 12348 x^{2} + 21609 $ 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 $\Q_{ 73 }$ to precision 11.
Roots:
$r_{ 1 }$ $=$ $ 3 + 66\cdot 73 + 3\cdot 73^{2} + 32\cdot 73^{3} + 60\cdot 73^{4} + 65\cdot 73^{5} + 29\cdot 73^{6} + 10\cdot 73^{7} + 24\cdot 73^{8} + 19\cdot 73^{9} + 21\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 2 }$ $=$ $ 11 + 48\cdot 73 + 43\cdot 73^{2} + 45\cdot 73^{3} + 46\cdot 73^{4} + 15\cdot 73^{5} + 51\cdot 73^{6} + 50\cdot 73^{7} + 19\cdot 73^{8} + 30\cdot 73^{9} + 5\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 3 }$ $=$ $ 20 + 32\cdot 73 + 36\cdot 73^{2} + 44\cdot 73^{3} + 2\cdot 73^{4} + 26\cdot 73^{5} + 52\cdot 73^{6} + 35\cdot 73^{7} + 53\cdot 73^{8} + 71\cdot 73^{9} +O\left(73^{ 11 }\right)$
$r_{ 4 }$ $=$ $ 24 + 31\cdot 73 + 53\cdot 73^{2} + 45\cdot 73^{3} + 33\cdot 73^{4} + 34\cdot 73^{5} + 61\cdot 73^{6} + 44\cdot 73^{7} + 73^{8} + 62\cdot 73^{9} + 27\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 5 }$ $=$ $ 49 + 41\cdot 73 + 19\cdot 73^{2} + 27\cdot 73^{3} + 39\cdot 73^{4} + 38\cdot 73^{5} + 11\cdot 73^{6} + 28\cdot 73^{7} + 71\cdot 73^{8} + 10\cdot 73^{9} + 45\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 6 }$ $=$ $ 53 + 40\cdot 73 + 36\cdot 73^{2} + 28\cdot 73^{3} + 70\cdot 73^{4} + 46\cdot 73^{5} + 20\cdot 73^{6} + 37\cdot 73^{7} + 19\cdot 73^{8} + 73^{9} + 72\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 7 }$ $=$ $ 62 + 24\cdot 73 + 29\cdot 73^{2} + 27\cdot 73^{3} + 26\cdot 73^{4} + 57\cdot 73^{5} + 21\cdot 73^{6} + 22\cdot 73^{7} + 53\cdot 73^{8} + 42\cdot 73^{9} + 67\cdot 73^{10} +O\left(73^{ 11 }\right)$
$r_{ 8 }$ $=$ $ 70 + 6\cdot 73 + 69\cdot 73^{2} + 40\cdot 73^{3} + 12\cdot 73^{4} + 7\cdot 73^{5} + 43\cdot 73^{6} + 62\cdot 73^{7} + 48\cdot 73^{8} + 53\cdot 73^{9} + 51\cdot 73^{10} +O\left(73^{ 11 }\right)$

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

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

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$$4$$(1,7,8,2)(3,4,6,5)$$0$
$2$$4$$(1,4,8,5)(2,6,7,3)$$0$
$2$$4$$(1,3,8,6)(2,4,7,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.