Properties

Label 1.171.6t1.c.b
Dimension $1$
Group $C_6$
Conductor $171$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(171\)\(\medspace = 3^{2} \cdot 19 \)
Artin field: Galois closure of 6.0.16245685539.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{171}(31,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - 95x^{3} + 684x^{2} - 570x + 2375 \) Copy content 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 $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a + 14 + \left(11 a + 27\right)\cdot 29 + \left(14 a + 19\right)\cdot 29^{2} + \left(15 a + 27\right)\cdot 29^{3} + \left(7 a + 11\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 a + \left(4 a + 23\right)\cdot 29 + \left(18 a + 23\right)\cdot 29^{2} + \left(19 a + 2\right)\cdot 29^{3} + \left(15 a + 11\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 17 a + 2 + \left(24 a + 4\right)\cdot 29 + \left(10 a + 23\right)\cdot 29^{2} + \left(9 a + 24\right)\cdot 29^{3} + \left(13 a + 11\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 28 a + 19 + \left(17 a + 23\right)\cdot 29 + \left(14 a + 22\right)\cdot 29^{2} + \left(13 a + 3\right)\cdot 29^{3} + \left(21 a + 5\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 13 a + 8 + \left(15 a + 1\right)\cdot 29 + \left(3 a + 12\right)\cdot 29^{2} + 6 a\cdot 29^{3} + \left(23 a + 12\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 16 a + 15 + \left(13 a + 7\right)\cdot 29 + \left(25 a + 14\right)\cdot 29^{2} + \left(22 a + 27\right)\cdot 29^{3} + \left(5 a + 5\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.