Properties

Label 1.2e2_7.6t1.2c2
Dimension 1
Group $C_6$
Conductor $ 2^{2} \cdot 7 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$28= 2^{2} \cdot 7 $
Artin number field: Splitting field of $f= x^{6} + 5 x^{4} + 6 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{28}(23,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots: \[ \begin{aligned} r_{ 1 } &= 32 a + 27 + \left(29 a + 22\right)\cdot 43 + \left(10 a + 9\right)\cdot 43^{2} + \left(18 a + 39\right)\cdot 43^{3} + \left(27 a + 16\right)\cdot 43^{4} +O\left(43^{ 5 }\right) \\ r_{ 2 } &= 10 a + 38 + \left(19 a + 16\right)\cdot 43 + \left(38 a + 33\right)\cdot 43^{2} + \left(15 a + 32\right)\cdot 43^{3} + \left(6 a + 4\right)\cdot 43^{4} +O\left(43^{ 5 }\right) \\ r_{ 3 } &= 26 a + 30 + \left(35 a + 16\right)\cdot 43 + \left(36 a + 42\right)\cdot 43^{2} + \left(28 a + 3\right)\cdot 43^{3} + \left(4 a + 12\right)\cdot 43^{4} +O\left(43^{ 5 }\right) \\ r_{ 4 } &= 11 a + 16 + \left(13 a + 20\right)\cdot 43 + \left(32 a + 33\right)\cdot 43^{2} + \left(24 a + 3\right)\cdot 43^{3} + \left(15 a + 26\right)\cdot 43^{4} +O\left(43^{ 5 }\right) \\ r_{ 5 } &= 33 a + 5 + \left(23 a + 26\right)\cdot 43 + \left(4 a + 9\right)\cdot 43^{2} + \left(27 a + 10\right)\cdot 43^{3} + \left(36 a + 38\right)\cdot 43^{4} +O\left(43^{ 5 }\right) \\ r_{ 6 } &= 17 a + 13 + \left(7 a + 26\right)\cdot 43 + 6 a\cdot 43^{2} + \left(14 a + 39\right)\cdot 43^{3} + \left(38 a + 30\right)\cdot 43^{4} +O\left(43^{ 5 }\right) \\ \end{aligned}\]

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

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

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$$3$$(1,5,6)(2,3,4)$$-\zeta_{3} - 1$
$1$$3$$(1,6,5)(2,4,3)$$\zeta_{3}$
$1$$6$$(1,3,5,4,6,2)$$-\zeta_{3}$
$1$$6$$(1,2,6,4,5,3)$$\zeta_{3} + 1$
The blue line marks the conjugacy class containing complex conjugation.