Properties

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

Related objects

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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} - 3 x^{5} - 12 x^{4} + 27 x^{3} + 21 x^{2} - 48 x + 17 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{117}(103,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 16 a + 12 + \left(14 a + 1\right)\cdot 19 + \left(6 a + 12\right)\cdot 19^{2} + \left(a + 1\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 16 a + 14 + \left(14 a + 14\right)\cdot 19 + \left(6 a + 9\right)\cdot 19^{2} + 14\cdot 19^{3} + \left(a + 15\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 3 a + 11 + \left(4 a + 13\right)\cdot 19 + \left(12 a + 1\right)\cdot 19^{2} + \left(18 a + 8\right)\cdot 19^{3} + \left(17 a + 16\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 3 a + 9 + 4 a\cdot 19 + \left(12 a + 4\right)\cdot 19^{2} + \left(18 a + 13\right)\cdot 19^{3} + \left(17 a + 1\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 16 a + 18 + \left(14 a + 13\right)\cdot 19 + \left(6 a + 18\right)\cdot 19^{2} + 3\cdot 19^{3} + \left(a + 1\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 3 a + 15 + \left(4 a + 12\right)\cdot 19 + \left(12 a + 10\right)\cdot 19^{2} + \left(18 a + 16\right)\cdot 19^{3} + \left(17 a + 1\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

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

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

Character values on conjugacy classes

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