Properties

Label 2.13e2_29e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 13^{2} \cdot 29^{2}$
Root number -1
Frobenius-Schur indicator -1

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$142129= 13^{2} \cdot 29^{2} $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 72 x^{6} - 592 x^{5} + 3519 x^{4} - 9968 x^{3} + 23522 x^{2} - 66393 x + 114383 $ 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 $\Q_{ 103 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 5 + 42\cdot 103 + 20\cdot 103^{2} + 95\cdot 103^{3} + 60\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 14 + 64\cdot 103 + 79\cdot 103^{2} + 33\cdot 103^{3} + 40\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 15 + 2\cdot 103 + 101\cdot 103^{2} + 77\cdot 103^{3} + 7\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 17 + 87\cdot 103 + 96\cdot 103^{2} + 3\cdot 103^{3} + 44\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 25 + 82\cdot 103 + 36\cdot 103^{2} + 16\cdot 103^{3} + 36\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 58 + 66\cdot 103 + 10\cdot 103^{2} + 103^{3} + 78\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 88 + 31\cdot 103 + 14\cdot 103^{2} + 40\cdot 103^{3} + 31\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 90 + 35\cdot 103 + 52\cdot 103^{2} + 40\cdot 103^{3} + 10\cdot 103^{4} +O\left(103^{ 5 }\right)$

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

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

Character values on conjugacy classes

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