Properties

Label 2.3e2_5e2_23e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 3^{2} \cdot 5^{2} \cdot 23^{2}$
Root number 1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$119025= 3^{2} \cdot 5^{2} \cdot 23^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 112 x^{6} + 95 x^{5} + 2881 x^{4} + 835 x^{3} - 16858 x^{2} - 25817 x - 10769 $ 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 $\Q_{ 89 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 59\cdot 89 + 83\cdot 89^{2} + 7\cdot 89^{3} + 74\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 11 + 17\cdot 89^{2} + 41\cdot 89^{3} + 22\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 23 + 68\cdot 89 + 66\cdot 89^{2} + 22\cdot 89^{3} + 59\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 46 + 36\cdot 89 + 72\cdot 89^{2} + 19\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 51 + 79\cdot 89 + 36\cdot 89^{2} + 81\cdot 89^{3} + 87\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 68 + 52\cdot 89 + 70\cdot 89^{2} + 52\cdot 89^{3} + 54\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 78 + 32\cdot 89 + 11\cdot 89^{2} + 52\cdot 89^{3} + 16\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 80 + 26\cdot 89 + 86\cdot 89^{2} + 7\cdot 89^{3} + 22\cdot 89^{4} +O\left(89^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,2)(3,4)(5,8)(6,7)$$-2$
$2$$4$$(1,7,2,6)(3,5,4,8)$$0$
$2$$4$$(1,5,2,8)(3,6,4,7)$$0$
$2$$4$$(1,3,2,4)(5,7,8,6)$$0$
The blue line marks the conjugacy class containing complex conjugation.