Properties

Label 2.3e2_7e2_23e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 3^{2} \cdot 7^{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:$233289= 3^{2} \cdot 7^{2} \cdot 23^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 136 x^{6} - 94 x^{5} + 5029 x^{4} + 6616 x^{3} - 37504 x^{2} + 14104 x + 1360 $ 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 $\Q_{ 17 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 2\cdot 17 + 4\cdot 17^{2} + 4\cdot 17^{3} + 9\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 1 + 12\cdot 17 + 5\cdot 17^{2} + 3\cdot 17^{3} + 12\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 2 + 13\cdot 17 + 2\cdot 17^{2} + 4\cdot 17^{3} + 2\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 3 + 5\cdot 17 + 4\cdot 17^{2} + 14\cdot 17^{3} + 4\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 4 + 8\cdot 17 + 15\cdot 17^{2} + 13\cdot 17^{3} + 12\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 5 + 14\cdot 17 + 16\cdot 17^{2} + 5\cdot 17^{3} +O\left(17^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 7 + 15\cdot 17 + 13\cdot 17^{2} + 8\cdot 17^{3} + 13\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 13 + 14\cdot 17 + 4\cdot 17^{2} + 13\cdot 17^{3} + 12\cdot 17^{4} +O\left(17^{ 5 }\right)$

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

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

Character values on conjugacy classes

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