Properties

Label 1.3_7.6t1.2c2
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} + 3 x^{4} + 5 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{21}(2,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 9 a + 12 + \left(16 a + 27\right)\cdot 29 + \left(22 a + 9\right)\cdot 29^{2} + \left(21 a + 22\right)\cdot 29^{3} + 10\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 a + 24 + \left(13 a + 19\right)\cdot 29 + \left(13 a + 8\right)\cdot 29^{2} + \left(10 a + 24\right)\cdot 29^{3} + \left(12 a + 16\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 13 a + 24 + \left(13 a + 23\right)\cdot 29 + \left(21 a + 18\right)\cdot 29^{2} + \left(21 a + 6\right)\cdot 29^{3} + \left(28 a + 25\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 20 a + 28 + \left(12 a + 12\right)\cdot 29 + \left(6 a + 19\right)\cdot 29^{2} + \left(7 a + 21\right)\cdot 29^{3} + \left(28 a + 21\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 11 a + 27 + \left(15 a + 11\right)\cdot 29 + \left(15 a + 4\right)\cdot 29^{2} + \left(18 a + 5\right)\cdot 29^{3} + \left(16 a + 10\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 16 a + 2 + \left(15 a + 20\right)\cdot 29 + \left(7 a + 25\right)\cdot 29^{2} + \left(7 a + 6\right)\cdot 29^{3} + 2\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

Character values on conjugacy classes

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