Properties

Label 1.63.6t1.h.a
Dimension $1$
Group $C_6$
Conductor $63$
Root number not computed
Indicator $0$

Related objects

Learn more about

Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(63\)\(\medspace = 3^{2} \cdot 7 \)
Artin number field: Galois closure of 6.6.6751269.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{63}(20,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{6} - 12 x^{4} - 5 x^{3} + 36 x^{2} + 30 x + 1 $.

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $

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

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

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

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

The blue line marks the conjugacy class containing complex conjugation.