Properties

Label 1.273.6t1.f.b
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: 6.6.3438544473.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{273}(251,\cdot)\)
Projective image: C_1
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} - 70 x^{4} + 69 x^{3} + 786 x^{2} - 68 x - 1301\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \(x^{2} + 45 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 26 a + 2 + \left(33 a + 32\right)\cdot 47 + \left(5 a + 24\right)\cdot 47^{2} + \left(44 a + 14\right)\cdot 47^{3} + \left(13 a + 35\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 21 a + 7 + \left(13 a + 26\right)\cdot 47 + \left(41 a + 2\right)\cdot 47^{2} + \left(2 a + 3\right)\cdot 47^{3} + \left(33 a + 19\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 46 a + 8 + \left(23 a + 17\right)\cdot 47 + \left(41 a + 37\right)\cdot 47^{2} + \left(21 a + 14\right)\cdot 47^{3} + \left(13 a + 3\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( a + 6 + \left(23 a + 19\right)\cdot 47 + \left(5 a + 2\right)\cdot 47^{2} + \left(25 a + 17\right)\cdot 47^{3} + \left(33 a + 8\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 14 a + 22 + \left(29 a + 24\right)\cdot 47 + \left(4 a + 23\right)\cdot 47^{2} + \left(21 a + 3\right)\cdot 47^{3} + \left(9 a + 15\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 33 a + 3 + \left(17 a + 22\right)\cdot 47 + \left(42 a + 3\right)\cdot 47^{2} + \left(25 a + 41\right)\cdot 47^{3} + \left(37 a + 12\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display

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

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

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$$3$$(1,3,6)(2,4,5)$$-\zeta_{3} - 1$
$1$$3$$(1,6,3)(2,5,4)$$\zeta_{3}$
$1$$6$$(1,4,6,2,3,5)$$\zeta_{3} + 1$
$1$$6$$(1,5,3,2,6,4)$$-\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.