Properties

Label 2.3e2_7e2_23e2.8t5.2c1
Dimension 2
Group $Q_8$
Conductor $ 3^{2} \cdot 7^{2} \cdot 23^{2}$
Root number -1
Frobenius-Schur indicator -1

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$233289= 3^{2} \cdot 7^{2} \cdot 23^{2} $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 109 x^{6} + 138 x^{5} + 3801 x^{4} + 13938 x^{3} + 54538 x^{2} + 67350 x + 58153 $ 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:
$r_{ 1 }$ $=$ $ 1 + 29\cdot 89^{2} + 7\cdot 89^{3} + 46\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 10 + 72\cdot 89 + 71\cdot 89^{2} + 47\cdot 89^{3} + 20\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 22 + 79\cdot 89 + 70\cdot 89^{2} + 39\cdot 89^{3} + 39\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 27 + 79\cdot 89 + 14\cdot 89^{2} + 2\cdot 89^{3} + 29\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 36 + 83\cdot 89 + 31\cdot 89^{2} + 62\cdot 89^{3} + 3\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 48 + 23\cdot 89 + 17\cdot 89^{2} + 50\cdot 89^{3} + 77\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 53 + 26\cdot 89 + 55\cdot 89^{2} + 7\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 73 + 80\cdot 89 + 64\cdot 89^{2} + 49\cdot 89^{3} + 6\cdot 89^{4} +O\left(89^{ 5 }\right)$

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

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

Character values on conjugacy classes

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