Properties

Label 1.180.6t1.b.a
Dimension $1$
Group $C_6$
Conductor $180$
Root number not computed
Indicator $0$

Related objects

Learn more about

Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Artin field: 6.0.52488000.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{180}(79,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} + 9 x^{4} - 2 x^{3} + 84 x^{2} + 36 x + 321\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \(x^{2} + 16 x + 3\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.