Properties

Label 2.2e8_3e2_7e2.8t5.3c1
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} + 1260 x^{4} + 5292 x^{2} + 441 $ 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 $\Q_{ 61 }$ to precision 12.
Roots:
$r_{ 1 }$ $=$ $ 5 + 61 + 25\cdot 61^{2} + 16\cdot 61^{3} + 19\cdot 61^{4} + 55\cdot 61^{5} + 57\cdot 61^{6} + 61^{7} + 8\cdot 61^{8} + 38\cdot 61^{9} + 8\cdot 61^{10} + 5\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 2 }$ $=$ $ 7 + 23\cdot 61 + 3\cdot 61^{2} + 43\cdot 61^{3} + 2\cdot 61^{4} + 37\cdot 61^{5} + 23\cdot 61^{6} + 14\cdot 61^{7} + 39\cdot 61^{8} + 8\cdot 61^{9} + 16\cdot 61^{10} + 41\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 3 }$ $=$ $ 17 + 11\cdot 61 + 50\cdot 61^{2} + 11\cdot 61^{3} + 49\cdot 61^{4} + 27\cdot 61^{5} + 61^{6} + 29\cdot 61^{7} + 23\cdot 61^{8} + 48\cdot 61^{9} + 43\cdot 61^{10} + 30\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 4 }$ $=$ $ 23 + 40\cdot 61 + 61^{2} + 9\cdot 61^{3} + 48\cdot 61^{4} + 3\cdot 61^{5} + 35\cdot 61^{6} + 30\cdot 61^{7} + 41\cdot 61^{8} + 3\cdot 61^{9} + 12\cdot 61^{10} + 3\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 5 }$ $=$ $ 38 + 20\cdot 61 + 59\cdot 61^{2} + 51\cdot 61^{3} + 12\cdot 61^{4} + 57\cdot 61^{5} + 25\cdot 61^{6} + 30\cdot 61^{7} + 19\cdot 61^{8} + 57\cdot 61^{9} + 48\cdot 61^{10} + 57\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 6 }$ $=$ $ 44 + 49\cdot 61 + 10\cdot 61^{2} + 49\cdot 61^{3} + 11\cdot 61^{4} + 33\cdot 61^{5} + 59\cdot 61^{6} + 31\cdot 61^{7} + 37\cdot 61^{8} + 12\cdot 61^{9} + 17\cdot 61^{10} + 30\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 7 }$ $=$ $ 54 + 37\cdot 61 + 57\cdot 61^{2} + 17\cdot 61^{3} + 58\cdot 61^{4} + 23\cdot 61^{5} + 37\cdot 61^{6} + 46\cdot 61^{7} + 21\cdot 61^{8} + 52\cdot 61^{9} + 44\cdot 61^{10} + 19\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 8 }$ $=$ $ 56 + 59\cdot 61 + 35\cdot 61^{2} + 44\cdot 61^{3} + 41\cdot 61^{4} + 5\cdot 61^{5} + 3\cdot 61^{6} + 59\cdot 61^{7} + 52\cdot 61^{8} + 22\cdot 61^{9} + 52\cdot 61^{10} + 55\cdot 61^{11} +O\left(61^{ 12 }\right)$

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

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

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