Properties

Label 1.43.6t1.a.b
Dimension $1$
Group $C_6$
Conductor $43$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(43\)
Artin field: Galois closure of 6.0.147008443.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{43}(37,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} + 4x^{4} + 23x^{3} + 67x^{2} + 50x + 44 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: \( x^{2} + 101x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 64 a + 3 + \left(85 a + 39\right)\cdot 113 + \left(53 a + 61\right)\cdot 113^{2} + \left(66 a + 56\right)\cdot 113^{3} + \left(37 a + 13\right)\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 112 a + 89 + \left(99 a + 25\right)\cdot 113 + \left(a + 87\right)\cdot 113^{2} + \left(39 a + 52\right)\cdot 113^{3} + \left(101 a + 101\right)\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 74 a + 47 + \left(74 a + 24\right)\cdot 113 + \left(62 a + 6\right)\cdot 113^{2} + \left(35 a + 64\right)\cdot 113^{3} + \left(94 a + 81\right)\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a + 77 + \left(13 a + 96\right)\cdot 113 + \left(111 a + 9\right)\cdot 113^{2} + \left(73 a + 67\right)\cdot 113^{3} + \left(11 a + 35\right)\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 39 a + 31 + \left(38 a + 55\right)\cdot 113 + \left(50 a + 5\right)\cdot 113^{2} + \left(77 a + 89\right)\cdot 113^{3} + \left(18 a + 47\right)\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 49 a + 93 + \left(27 a + 97\right)\cdot 113 + \left(59 a + 55\right)\cdot 113^{2} + \left(46 a + 9\right)\cdot 113^{3} + \left(75 a + 59\right)\cdot 113^{4} +O(113^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,6)(2,4)(3,5)$$-1$
$1$$3$$(1,5,4)(2,6,3)$$-\zeta_{3} - 1$
$1$$3$$(1,4,5)(2,3,6)$$\zeta_{3}$
$1$$6$$(1,2,5,6,4,3)$$-\zeta_{3}$
$1$$6$$(1,3,4,6,5,2)$$\zeta_{3} + 1$

The blue line marks the conjugacy class containing complex conjugation.