Properties

Label 1.273.6t1.i.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: 6.6.12960667629.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{273}(269,\cdot)\)
Projective image: C_1
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} - 62 x^{4} + 55 x^{3} + 631 x^{2} + 48 x - 944\)  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:
$r_{ 1 }$ $=$ \( 4 a + 7 + \left(19 a + 18\right)\cdot 23 + \left(21 a + 17\right)\cdot 23^{2} + \left(5 a + 4\right)\cdot 23^{3} + \left(2 a + 15\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 22 + \left(6 a + 19\right)\cdot 23 + 9\cdot 23^{2} + \left(12 a + 14\right)\cdot 23^{3} + \left(8 a + 10\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 16 a + 13 + \left(16 a + 2\right)\cdot 23 + \left(22 a + 4\right)\cdot 23^{2} + \left(10 a + 15\right)\cdot 23^{3} + \left(14 a + 15\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 10 a + 8 + \left(14 a + 1\right)\cdot 23 + \left(12 a + 15\right)\cdot 23^{2} + \left(21 a + 4\right)\cdot 23^{3} + \left(8 a + 20\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 19 a + 15 + \left(3 a + 6\right)\cdot 23 + \left(a + 19\right)\cdot 23^{2} + \left(17 a + 17\right)\cdot 23^{3} + \left(20 a + 13\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 13 a + 5 + \left(8 a + 20\right)\cdot 23 + \left(10 a + 2\right)\cdot 23^{2} + \left(a + 12\right)\cdot 23^{3} + \left(14 a + 16\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.