Properties

Label 2.2e8_3e2_11e2.8t5.11c1
Dimension 2
Group $Q_8$
Conductor $ 2^{8} \cdot 3^{2} \cdot 11^{2}$
Root number 1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$278784= 2^{8} \cdot 3^{2} \cdot 11^{2} $
Artin number field: Splitting field of $f= x^{8} - 132 x^{6} + 5940 x^{4} - 100188 x^{2} + 393129 $ 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 $\Q_{ 97 }$ to precision 11.
Roots:
$r_{ 1 }$ $=$ $ 9 + 20\cdot 97 + 94\cdot 97^{2} + 51\cdot 97^{3} + 43\cdot 97^{4} + 91\cdot 97^{5} + 84\cdot 97^{6} + 67\cdot 97^{7} + 56\cdot 97^{8} + 79\cdot 97^{9} + 42\cdot 97^{10} +O\left(97^{ 11 }\right)$
$r_{ 2 }$ $=$ $ 11 + 37\cdot 97 + 39\cdot 97^{2} + 86\cdot 97^{3} + 13\cdot 97^{4} + 72\cdot 97^{5} + 9\cdot 97^{6} + 11\cdot 97^{7} + 97^{8} + 32\cdot 97^{9} + 94\cdot 97^{10} +O\left(97^{ 11 }\right)$
$r_{ 3 }$ $=$ $ 19 + 37\cdot 97 + 45\cdot 97^{2} + 8\cdot 97^{3} + 30\cdot 97^{4} + 44\cdot 97^{5} + 93\cdot 97^{6} + 67\cdot 97^{7} + 51\cdot 97^{8} + 70\cdot 97^{9} + 88\cdot 97^{10} +O\left(97^{ 11 }\right)$
$r_{ 4 }$ $=$ $ 32 + 61\cdot 97 + 95\cdot 97^{2} + 39\cdot 97^{3} + 59\cdot 97^{4} + 60\cdot 97^{5} + 39\cdot 97^{7} + 71\cdot 97^{8} + 41\cdot 97^{9} + 31\cdot 97^{10} +O\left(97^{ 11 }\right)$
$r_{ 5 }$ $=$ $ 65 + 35\cdot 97 + 97^{2} + 57\cdot 97^{3} + 37\cdot 97^{4} + 36\cdot 97^{5} + 96\cdot 97^{6} + 57\cdot 97^{7} + 25\cdot 97^{8} + 55\cdot 97^{9} + 65\cdot 97^{10} +O\left(97^{ 11 }\right)$
$r_{ 6 }$ $=$ $ 78 + 59\cdot 97 + 51\cdot 97^{2} + 88\cdot 97^{3} + 66\cdot 97^{4} + 52\cdot 97^{5} + 3\cdot 97^{6} + 29\cdot 97^{7} + 45\cdot 97^{8} + 26\cdot 97^{9} + 8\cdot 97^{10} +O\left(97^{ 11 }\right)$
$r_{ 7 }$ $=$ $ 86 + 59\cdot 97 + 57\cdot 97^{2} + 10\cdot 97^{3} + 83\cdot 97^{4} + 24\cdot 97^{5} + 87\cdot 97^{6} + 85\cdot 97^{7} + 95\cdot 97^{8} + 64\cdot 97^{9} + 2\cdot 97^{10} +O\left(97^{ 11 }\right)$
$r_{ 8 }$ $=$ $ 88 + 76\cdot 97 + 2\cdot 97^{2} + 45\cdot 97^{3} + 53\cdot 97^{4} + 5\cdot 97^{5} + 12\cdot 97^{6} + 29\cdot 97^{7} + 40\cdot 97^{8} + 17\cdot 97^{9} + 54\cdot 97^{10} +O\left(97^{ 11 }\right)$

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

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

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