Properties

Label 1.3_7.6t1.1c2
Dimension 1
Group $C_6$
Conductor $ 3 \cdot 7 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$21= 3 \cdot 7 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{21}(5,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 11 a + 9 + \left(5 a + 3\right)\cdot 13 + \left(9 a + 7\right)\cdot 13^{2} + \left(3 a + 5\right)\cdot 13^{3} + \left(2 a + 6\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 10 a + 4 + \left(9 a + 5\right)\cdot 13 + 12 a\cdot 13^{2} + 7 a\cdot 13^{3} + \left(12 a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 2 a + 7 + \left(7 a + 11\right)\cdot 13 + \left(3 a + 10\right)\cdot 13^{2} + \left(9 a + 12\right)\cdot 13^{3} + \left(10 a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 6 a + \left(12 a + 10\right)\cdot 13 + \left(8 a + 3\right)\cdot 13^{2} + \left(2 a + 9\right)\cdot 13^{3} + \left(9 a + 10\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 7 a + 6 + 3\cdot 13 + 4 a\cdot 13^{2} + \left(10 a + 3\right)\cdot 13^{3} + \left(3 a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 3 a + 1 + \left(3 a + 5\right)\cdot 13 + 3\cdot 13^{2} + \left(5 a + 8\right)\cdot 13^{3} + 8\cdot 13^{4} +O\left(13^{ 5 }\right)$

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

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

Character values on conjugacy classes

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