Properties

Label 2.2e8_3e2_11e2.8t5.2c1
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 $\Q_{ 79 }$ to precision 12.
Roots:
$r_{ 1 }$ $=$ $ 15 + 74\cdot 79 + 7\cdot 79^{2} + 24\cdot 79^{3} + 60\cdot 79^{4} + 77\cdot 79^{5} + 62\cdot 79^{6} + 23\cdot 79^{7} + 49\cdot 79^{8} + 30\cdot 79^{9} + 10\cdot 79^{10} + 73\cdot 79^{11} +O\left(79^{ 12 }\right)$
$r_{ 2 }$ $=$ $ 26 + 53\cdot 79 + 71\cdot 79^{2} + 19\cdot 79^{3} + 42\cdot 79^{4} + 26\cdot 79^{5} + 75\cdot 79^{6} + 69\cdot 79^{7} + 66\cdot 79^{8} + 46\cdot 79^{9} + 12\cdot 79^{10} + 22\cdot 79^{11} +O\left(79^{ 12 }\right)$
$r_{ 3 }$ $=$ $ 29 + 66\cdot 79 + 62\cdot 79^{2} + 53\cdot 79^{3} + 61\cdot 79^{4} + 7\cdot 79^{5} + 79^{6} + 54\cdot 79^{7} + 49\cdot 79^{8} + 46\cdot 79^{9} + 30\cdot 79^{10} + 65\cdot 79^{11} +O\left(79^{ 12 }\right)$
$r_{ 4 }$ $=$ $ 38 + 52\cdot 79 + 58\cdot 79^{2} + 33\cdot 79^{3} + 32\cdot 79^{4} + 67\cdot 79^{5} + 37\cdot 79^{6} + 43\cdot 79^{7} + 34\cdot 79^{8} + 6\cdot 79^{9} + 30\cdot 79^{10} + 75\cdot 79^{11} +O\left(79^{ 12 }\right)$
$r_{ 5 }$ $=$ $ 41 + 26\cdot 79 + 20\cdot 79^{2} + 45\cdot 79^{3} + 46\cdot 79^{4} + 11\cdot 79^{5} + 41\cdot 79^{6} + 35\cdot 79^{7} + 44\cdot 79^{8} + 72\cdot 79^{9} + 48\cdot 79^{10} + 3\cdot 79^{11} +O\left(79^{ 12 }\right)$
$r_{ 6 }$ $=$ $ 50 + 12\cdot 79 + 16\cdot 79^{2} + 25\cdot 79^{3} + 17\cdot 79^{4} + 71\cdot 79^{5} + 77\cdot 79^{6} + 24\cdot 79^{7} + 29\cdot 79^{8} + 32\cdot 79^{9} + 48\cdot 79^{10} + 13\cdot 79^{11} +O\left(79^{ 12 }\right)$
$r_{ 7 }$ $=$ $ 53 + 25\cdot 79 + 7\cdot 79^{2} + 59\cdot 79^{3} + 36\cdot 79^{4} + 52\cdot 79^{5} + 3\cdot 79^{6} + 9\cdot 79^{7} + 12\cdot 79^{8} + 32\cdot 79^{9} + 66\cdot 79^{10} + 56\cdot 79^{11} +O\left(79^{ 12 }\right)$
$r_{ 8 }$ $=$ $ 64 + 4\cdot 79 + 71\cdot 79^{2} + 54\cdot 79^{3} + 18\cdot 79^{4} + 79^{5} + 16\cdot 79^{6} + 55\cdot 79^{7} + 29\cdot 79^{8} + 48\cdot 79^{9} + 68\cdot 79^{10} + 5\cdot 79^{11} +O\left(79^{ 12 }\right)$

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

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

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