Properties

Label 2.2e8_3e2_11e2.8t5.1c1
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} + 2772 x^{4} - 13068 x^{2} + 9801 $ 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 12.
Roots:
$r_{ 1 }$ $=$ $ 5 + 26\cdot 97 + 68\cdot 97^{2} + 41\cdot 97^{3} + 95\cdot 97^{4} + 9\cdot 97^{5} + 31\cdot 97^{6} + 40\cdot 97^{7} + 62\cdot 97^{8} + 13\cdot 97^{9} + 58\cdot 97^{10} + 46\cdot 97^{11} +O\left(97^{ 12 }\right)$
$r_{ 2 }$ $=$ $ 32 + 84\cdot 97 + 68\cdot 97^{2} + 2\cdot 97^{3} + 26\cdot 97^{4} + 57\cdot 97^{5} + 11\cdot 97^{6} + 30\cdot 97^{7} + 88\cdot 97^{8} + 93\cdot 97^{9} + 87\cdot 97^{10} + 50\cdot 97^{11} +O\left(97^{ 12 }\right)$
$r_{ 3 }$ $=$ $ 36 + 75\cdot 97 + 97^{2} + 16\cdot 97^{3} + 66\cdot 97^{4} + 19\cdot 97^{5} + 29\cdot 97^{6} + 72\cdot 97^{7} + 88\cdot 97^{8} + 80\cdot 97^{9} + 79\cdot 97^{10} + 71\cdot 97^{11} +O\left(97^{ 12 }\right)$
$r_{ 4 }$ $=$ $ 42 + 91\cdot 97 + 20\cdot 97^{2} + 43\cdot 97^{3} + 96\cdot 97^{4} + 61\cdot 97^{5} + 96\cdot 97^{6} + 80\cdot 97^{7} + 25\cdot 97^{8} + 58\cdot 97^{9} + 64\cdot 97^{10} + 37\cdot 97^{11} +O\left(97^{ 12 }\right)$
$r_{ 5 }$ $=$ $ 55 + 5\cdot 97 + 76\cdot 97^{2} + 53\cdot 97^{3} + 35\cdot 97^{5} + 16\cdot 97^{7} + 71\cdot 97^{8} + 38\cdot 97^{9} + 32\cdot 97^{10} + 59\cdot 97^{11} +O\left(97^{ 12 }\right)$
$r_{ 6 }$ $=$ $ 61 + 21\cdot 97 + 95\cdot 97^{2} + 80\cdot 97^{3} + 30\cdot 97^{4} + 77\cdot 97^{5} + 67\cdot 97^{6} + 24\cdot 97^{7} + 8\cdot 97^{8} + 16\cdot 97^{9} + 17\cdot 97^{10} + 25\cdot 97^{11} +O\left(97^{ 12 }\right)$
$r_{ 7 }$ $=$ $ 65 + 12\cdot 97 + 28\cdot 97^{2} + 94\cdot 97^{3} + 70\cdot 97^{4} + 39\cdot 97^{5} + 85\cdot 97^{6} + 66\cdot 97^{7} + 8\cdot 97^{8} + 3\cdot 97^{9} + 9\cdot 97^{10} + 46\cdot 97^{11} +O\left(97^{ 12 }\right)$
$r_{ 8 }$ $=$ $ 92 + 70\cdot 97 + 28\cdot 97^{2} + 55\cdot 97^{3} + 97^{4} + 87\cdot 97^{5} + 65\cdot 97^{6} + 56\cdot 97^{7} + 34\cdot 97^{8} + 83\cdot 97^{9} + 38\cdot 97^{10} + 50\cdot 97^{11} +O\left(97^{ 12 }\right)$

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

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

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