Properties

Label 1.111.6t1.a.a
Dimension $1$
Group $C_6$
Conductor $111$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(111\)\(\medspace = 3 \cdot 37 \)
Artin field: 6.0.1872286839.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{111}(101,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} + 22 x^{4} - 83 x^{3} + 52 x^{2} + 36 x + 184\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \(x^{2} + 7 x + 2\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.