Properties

Label 1.7.6t1.1c2
Dimension 1
Group $C_6$
Conductor $ 7 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

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

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:
$r_{ 1 }$ $=$ $ 6 a + 6 + \left(9 a + 3\right)\cdot 13 + \left(2 a + 5\right)\cdot 13^{2} + \left(3 a + 10\right)\cdot 13^{3} + \left(10 a + 2\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ a + 9 + \left(8 a + 9\right)\cdot 13 + 10 a\cdot 13^{2} + \left(a + 4\right)\cdot 13^{3} + \left(2 a + 7\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 4 a + 6 + \left(3 a + 1\right)\cdot 13 + \left(9 a + 6\right)\cdot 13^{2} + \left(2 a + 12\right)\cdot 13^{3} + \left(5 a + 10\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 9 a + 10 + 9 a\cdot 13 + \left(3 a + 12\right)\cdot 13^{2} + \left(10 a + 5\right)\cdot 13^{3} + 7 a\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 12 a + 10 + \left(4 a + 3\right)\cdot 13 + \left(2 a + 3\right)\cdot 13^{2} + \left(11 a + 8\right)\cdot 13^{3} + \left(10 a + 7\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 7 a + 12 + \left(3 a + 6\right)\cdot 13 + \left(10 a + 11\right)\cdot 13^{2} + \left(9 a + 10\right)\cdot 13^{3} + \left(2 a + 9\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$

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

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

Character values on conjugacy classes

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