Properties

Label 1.273.6t1.b.a
Dimension $1$
Group $C_6$
Conductor $273$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \)
Artin field: 6.6.168488679177.2
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{273}(17,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} - 83 x^{4} + 433 x^{3} - 566 x^{2} - 120 x + 64\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \(x^{2} + 21 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 a + 3 + \left(7 a + 9\right)\cdot 23 + \left(11 a + 22\right)\cdot 23^{2} + \left(9 a + 18\right)\cdot 23^{3} + \left(9 a + 9\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 15 a + 19 + \left(15 a + 15\right)\cdot 23 + \left(11 a + 14\right)\cdot 23^{2} + \left(13 a + 3\right)\cdot 23^{3} + \left(13 a + 19\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 10 a + 8 + \left(9 a + 6\right)\cdot 23 + \left(22 a + 14\right)\cdot 23^{2} + \left(10 a + 8\right)\cdot 23^{3} + \left(7 a + 16\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 13 a + 5 + \left(13 a + 15\right)\cdot 23 + 3\cdot 23^{2} + \left(12 a + 8\right)\cdot 23^{3} + \left(15 a + 20\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 6 a + \left(13 a + 1\right)\cdot 23 + \left(17 a + 19\right)\cdot 23^{2} + \left(20 a + 2\right)\cdot 23^{3} + \left(16 a + 18\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 17 a + 12 + \left(9 a + 21\right)\cdot 23 + \left(5 a + 17\right)\cdot 23^{2} + \left(2 a + 3\right)\cdot 23^{3} + \left(6 a + 8\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.