Properties

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

Related objects

Learn more

Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \)
Artin field: 6.6.996974433.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{273}(194,\cdot)\)
Projective image: C_1
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} - 69 x^{4} + 69 x^{3} + 1331 x^{2} - 1331 x - 5669\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \(x^{2} + 24 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 9 a + 3 + \left(23 a + 20\right)\cdot 29 + \left(18 a + 14\right)\cdot 29^{2} + 22\cdot 29^{3} + \left(17 a + 14\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 5 a + 22 + \left(20 a + 15\right)\cdot 29 + \left(2 a + 3\right)\cdot 29^{2} + \left(22 a + 26\right)\cdot 29^{3} + \left(6 a + 24\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 3 a + 20 + \left(13 a + 19\right)\cdot 29 + \left(9 a + 19\right)\cdot 29^{2} + 10\cdot 29^{3} + \left(3 a + 6\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 26 a + 6 + \left(15 a + 24\right)\cdot 29 + \left(19 a + 24\right)\cdot 29^{2} + \left(28 a + 2\right)\cdot 29^{3} + \left(25 a + 21\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 24 a + 18 + \left(8 a + 24\right)\cdot 29 + \left(26 a + 25\right)\cdot 29^{2} + \left(6 a + 17\right)\cdot 29^{3} + \left(22 a + 7\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 20 a + 19 + \left(5 a + 11\right)\cdot 29 + \left(10 a + 27\right)\cdot 29^{2} + \left(28 a + 6\right)\cdot 29^{3} + \left(11 a + 12\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.