Properties

Label 1.252.6t1.j.b
Dimension $1$
Group $C_6$
Conductor $252$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \)
Artin field: 6.0.432081216.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{252}(167,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} + 42 x^{4} + 441 x^{2} + 1029\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \(x^{2} + 49 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 31 a + 44 + \left(36 a + 21\right)\cdot 53 + \left(51 a + 47\right)\cdot 53^{2} + \left(22 a + 32\right)\cdot 53^{3} + \left(32 a + 52\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 25 a + 3 + \left(15 a + 8\right)\cdot 53 + \left(49 a + 15\right)\cdot 53^{2} + \left(22 a + 5\right)\cdot 53^{3} + \left(38 a + 14\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 47 a + 12 + \left(31 a + 39\right)\cdot 53 + \left(50 a + 20\right)\cdot 53^{2} + \left(52 a + 25\right)\cdot 53^{3} + \left(5 a + 14\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 22 a + 9 + \left(16 a + 31\right)\cdot 53 + \left(a + 5\right)\cdot 53^{2} + \left(30 a + 20\right)\cdot 53^{3} + 20 a\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 28 a + 50 + \left(37 a + 44\right)\cdot 53 + \left(3 a + 37\right)\cdot 53^{2} + \left(30 a + 47\right)\cdot 53^{3} + \left(14 a + 38\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 6 a + 41 + \left(21 a + 13\right)\cdot 53 + \left(2 a + 32\right)\cdot 53^{2} + 27\cdot 53^{3} + \left(47 a + 38\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.