Properties

Label 1.171.6t1.g.a
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 6.0.2565108243.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{171}(83,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - 152x^{3} + 6859 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 21 a + 9 + \left(a + 12\right)\cdot 23 + \left(12 a + 2\right)\cdot 23^{2} + \left(14 a + 18\right)\cdot 23^{3} + \left(21 a + 1\right)\cdot 23^{4} + \left(15 a + 4\right)\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 17 a + \left(13 a + 13\right)\cdot 23 + \left(9 a + 15\right)\cdot 23^{2} + \left(21 a + 21\right)\cdot 23^{3} + \left(7 a + 18\right)\cdot 23^{4} + \left(14 a + 11\right)\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 6 a + 11 + 9 a\cdot 23 + \left(13 a + 21\right)\cdot 23^{2} + \left(a + 8\right)\cdot 23^{3} + \left(15 a + 13\right)\cdot 23^{4} + \left(8 a + 9\right)\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 2 a + 5 + \left(21 a + 18\right)\cdot 23 + \left(10 a + 1\right)\cdot 23^{2} + \left(8 a + 12\right)\cdot 23^{3} + \left(a + 7\right)\cdot 23^{4} + \left(7 a + 14\right)\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 19 a + 3 + \left(11 a + 10\right)\cdot 23 + \left(20 a + 22\right)\cdot 23^{2} + \left(6 a + 18\right)\cdot 23^{3} + \left(9 a + 7\right)\cdot 23^{4} + \left(21 a + 9\right)\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 4 a + 18 + \left(11 a + 14\right)\cdot 23 + \left(2 a + 5\right)\cdot 23^{2} + \left(16 a + 12\right)\cdot 23^{3} + \left(13 a + 19\right)\cdot 23^{4} + \left(a + 19\right)\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.