Properties

Label 1.3e2_13.6t1.3c1
Dimension 1
Group $C_6$
Conductor $ 3^{2} \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$117= 3^{2} \cdot 13 $
Artin number field: Splitting field of $f= x^{6} + 18 x^{4} - 10 x^{3} + 81 x^{2} - 90 x + 181 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{117}(77,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{2} + 16 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 12 a + 4 + \left(10 a + 15\right)\cdot 17 + \left(16 a + 12\right)\cdot 17^{2} + \left(10 a + 1\right)\cdot 17^{3} + \left(11 a + 2\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 15 a + 6 + \left(10 a + 8\right)\cdot 17 + \left(10 a + 11\right)\cdot 17^{2} + \left(10 a + 12\right)\cdot 17^{3} + \left(14 a + 4\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 2 a + 4 + \left(6 a + 4\right)\cdot 17 + \left(6 a + 11\right)\cdot 17^{2} + \left(6 a + 12\right)\cdot 17^{3} + \left(2 a + 8\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 3 a + 9 + 14\cdot 17 + \left(11 a + 9\right)\cdot 17^{2} + \left(16 a + 2\right)\cdot 17^{3} + \left(2 a + 6\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 14 a + 12 + \left(16 a + 11\right)\cdot 17 + \left(5 a + 3\right)\cdot 17^{2} + 8\cdot 17^{3} + \left(14 a + 9\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 5 a + 16 + \left(6 a + 13\right)\cdot 17 + 17^{2} + \left(6 a + 13\right)\cdot 17^{3} + \left(5 a + 2\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

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