Properties

Label 2.2e8_3e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 2^{8} \cdot 3^{2}$
Root number 1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$2304= 2^{8} \cdot 3^{2} $
Artin number field: Splitting field of $f= x^{8} - 12 x^{6} + 36 x^{4} - 36 x^{2} + 9 $ 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 $\Q_{ 23 }$ to precision 10.
Roots:
$r_{ 1 }$ $=$ $ 3 + 8\cdot 23 + 20\cdot 23^{2} + 8\cdot 23^{3} + 16\cdot 23^{4} + 17\cdot 23^{5} + 20\cdot 23^{6} + 15\cdot 23^{7} + 19\cdot 23^{8} + 6\cdot 23^{9} +O\left(23^{ 10 }\right)$
$r_{ 2 }$ $=$ $ 9 + 19\cdot 23 + 16\cdot 23^{2} + 4\cdot 23^{3} + 16\cdot 23^{4} + 14\cdot 23^{5} + 19\cdot 23^{6} + 3\cdot 23^{7} + 20\cdot 23^{8} + 22\cdot 23^{9} +O\left(23^{ 10 }\right)$
$r_{ 3 }$ $=$ $ 10 + 8\cdot 23 + 22\cdot 23^{2} + 17\cdot 23^{3} + 19\cdot 23^{4} + 5\cdot 23^{5} + 9\cdot 23^{6} + 15\cdot 23^{7} + 14\cdot 23^{8} + 17\cdot 23^{9} +O\left(23^{ 10 }\right)$
$r_{ 4 }$ $=$ $ 11 + 11\cdot 23 + 21\cdot 23^{2} + 10\cdot 23^{3} + 6\cdot 23^{4} + 20\cdot 23^{5} + 17\cdot 23^{6} + 3\cdot 23^{7} + 14\cdot 23^{8} + 19\cdot 23^{9} +O\left(23^{ 10 }\right)$
$r_{ 5 }$ $=$ $ 12 + 11\cdot 23 + 23^{2} + 12\cdot 23^{3} + 16\cdot 23^{4} + 2\cdot 23^{5} + 5\cdot 23^{6} + 19\cdot 23^{7} + 8\cdot 23^{8} + 3\cdot 23^{9} +O\left(23^{ 10 }\right)$
$r_{ 6 }$ $=$ $ 13 + 14\cdot 23 + 5\cdot 23^{3} + 3\cdot 23^{4} + 17\cdot 23^{5} + 13\cdot 23^{6} + 7\cdot 23^{7} + 8\cdot 23^{8} + 5\cdot 23^{9} +O\left(23^{ 10 }\right)$
$r_{ 7 }$ $=$ $ 14 + 3\cdot 23 + 6\cdot 23^{2} + 18\cdot 23^{3} + 6\cdot 23^{4} + 8\cdot 23^{5} + 3\cdot 23^{6} + 19\cdot 23^{7} + 2\cdot 23^{8} +O\left(23^{ 10 }\right)$
$r_{ 8 }$ $=$ $ 20 + 14\cdot 23 + 2\cdot 23^{2} + 14\cdot 23^{3} + 6\cdot 23^{4} + 5\cdot 23^{5} + 2\cdot 23^{6} + 7\cdot 23^{7} + 3\cdot 23^{8} + 16\cdot 23^{9} +O\left(23^{ 10 }\right)$

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

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

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