Properties

Label 2.2e8_11e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 2^{8} \cdot 11^{2}$
Root number 1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$30976= 2^{8} \cdot 11^{2} $
Artin number field: Splitting field of $f= x^{8} - 44 x^{6} + 308 x^{4} - 484 x^{2} + 121 $ 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 10.
Roots:
$r_{ 1 }$ $=$ $ 2 + 32\cdot 79 + 64\cdot 79^{2} + 77\cdot 79^{3} + 43\cdot 79^{4} + 47\cdot 79^{6} + 15\cdot 79^{7} + 32\cdot 79^{8} + 40\cdot 79^{9} +O\left(79^{ 10 }\right)$
$r_{ 2 }$ $=$ $ 14 + 46\cdot 79 + 52\cdot 79^{2} + 44\cdot 79^{3} + 53\cdot 79^{4} + 15\cdot 79^{5} + 20\cdot 79^{6} + 65\cdot 79^{7} + 62\cdot 79^{8} + 51\cdot 79^{9} +O\left(79^{ 10 }\right)$
$r_{ 3 }$ $=$ $ 16 + 54\cdot 79 + 64\cdot 79^{2} + 18\cdot 79^{3} + 27\cdot 79^{4} + 18\cdot 79^{5} + 43\cdot 79^{6} + 29\cdot 79^{7} + 61\cdot 79^{8} + 64\cdot 79^{9} +O\left(79^{ 10 }\right)$
$r_{ 4 }$ $=$ $ 33 + 61\cdot 79 + 76\cdot 79^{2} + 61\cdot 79^{3} + 45\cdot 79^{4} + 75\cdot 79^{5} + 37\cdot 79^{6} + 66\cdot 79^{7} + 45\cdot 79^{8} + 58\cdot 79^{9} +O\left(79^{ 10 }\right)$
$r_{ 5 }$ $=$ $ 46 + 17\cdot 79 + 2\cdot 79^{2} + 17\cdot 79^{3} + 33\cdot 79^{4} + 3\cdot 79^{5} + 41\cdot 79^{6} + 12\cdot 79^{7} + 33\cdot 79^{8} + 20\cdot 79^{9} +O\left(79^{ 10 }\right)$
$r_{ 6 }$ $=$ $ 63 + 24\cdot 79 + 14\cdot 79^{2} + 60\cdot 79^{3} + 51\cdot 79^{4} + 60\cdot 79^{5} + 35\cdot 79^{6} + 49\cdot 79^{7} + 17\cdot 79^{8} + 14\cdot 79^{9} +O\left(79^{ 10 }\right)$
$r_{ 7 }$ $=$ $ 65 + 32\cdot 79 + 26\cdot 79^{2} + 34\cdot 79^{3} + 25\cdot 79^{4} + 63\cdot 79^{5} + 58\cdot 79^{6} + 13\cdot 79^{7} + 16\cdot 79^{8} + 27\cdot 79^{9} +O\left(79^{ 10 }\right)$
$r_{ 8 }$ $=$ $ 77 + 46\cdot 79 + 14\cdot 79^{2} + 79^{3} + 35\cdot 79^{4} + 78\cdot 79^{5} + 31\cdot 79^{6} + 63\cdot 79^{7} + 46\cdot 79^{8} + 38\cdot 79^{9} +O\left(79^{ 10 }\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.