Properties

Label 1.287.6t1.a.b
Dimension $1$
Group $C_6$
Conductor $287$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(287\)\(\medspace = 7 \cdot 41 \)
Artin field: 6.6.165479321.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{287}(163,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} - 35 x^{4} + 23 x^{3} + 315 x^{2} - 121 x - 631\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{2} + 12 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 10 a + 12 + \left(12 a + 2\right)\cdot 13 + a\cdot 13^{2} + \left(7 a + 5\right)\cdot 13^{3} + \left(3 a + 3\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 10 a + 9 + \left(12 a + 4\right)\cdot 13 + \left(a + 1\right)\cdot 13^{2} + \left(7 a + 11\right)\cdot 13^{3} + \left(3 a + 12\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 3 a + 7 + 10\cdot 13 + \left(11 a + 3\right)\cdot 13^{2} + \left(5 a + 7\right)\cdot 13^{3} + \left(9 a + 11\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 10 a + 10 + \left(12 a + 7\right)\cdot 13 + \left(a + 1\right)\cdot 13^{2} + \left(7 a + 2\right)\cdot 13^{3} + \left(3 a + 2\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 3 a + 9 + 5\cdot 13 + \left(11 a + 2\right)\cdot 13^{2} + \left(5 a + 10\right)\cdot 13^{3} + \left(9 a + 12\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 3 a + 6 + 7\cdot 13 + \left(11 a + 3\right)\cdot 13^{2} + \left(5 a + 3\right)\cdot 13^{3} + \left(9 a + 9\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.