Properties

Label 1.455.6t1.b.b
Dimension $1$
Group $C_6$
Conductor $455$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(455\)\(\medspace = 5 \cdot 7 \cdot 13 \)
Artin field: 6.0.1224552875.2
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{455}(139,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} + 18 x^{4} - 7 x^{3} + 249 x^{2} - 230 x + 1535\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \(x^{2} + 29 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 4 a + 21 + \left(13 a + 15\right)\cdot 31 + \left(24 a + 19\right)\cdot 31^{2} + 20 a\cdot 31^{3} + \left(25 a + 18\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 27 a + 16 + \left(17 a + 20\right)\cdot 31 + \left(6 a + 17\right)\cdot 31^{2} + \left(10 a + 29\right)\cdot 31^{3} + \left(5 a + 9\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 27 a + 29 + \left(17 a + 6\right)\cdot 31 + \left(6 a + 24\right)\cdot 31^{2} + \left(10 a + 17\right)\cdot 31^{3} + \left(5 a + 17\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 4 a + 6 + \left(13 a + 30\right)\cdot 31 + \left(24 a + 6\right)\cdot 31^{2} + \left(20 a + 23\right)\cdot 31^{3} + \left(25 a + 18\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 4 a + 8 + \left(13 a + 29\right)\cdot 31 + \left(24 a + 12\right)\cdot 31^{2} + \left(20 a + 12\right)\cdot 31^{3} + \left(25 a + 10\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 27 a + 14 + \left(17 a + 21\right)\cdot 31 + \left(6 a + 11\right)\cdot 31^{2} + \left(10 a + 9\right)\cdot 31^{3} + \left(5 a + 18\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.