Properties

Label 1.129.6t1.a.b
Dimension $1$
Group $C_6$
Conductor $129$
Root number not computed
Indicator $0$

Related objects

Learn more

Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(129\)\(\medspace = 3 \cdot 43 \)
Artin field: 6.0.92307627.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{129}(122,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} + 15 x^{4} + 30 x^{3} + 188 x^{2} + 112 x + 64\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \(x^{2} + 38 x + 6\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 12 a + 27 + \left(39 a + 10\right)\cdot 41 + \left(8 a + 20\right)\cdot 41^{2} + \left(19 a + 18\right)\cdot 41^{3} + \left(21 a + 13\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 14 a + 29 + \left(2 a + 8\right)\cdot 41 + \left(4 a + 29\right)\cdot 41^{2} + \left(3 a + 19\right)\cdot 41^{3} + \left(3 a + 25\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 17 a + 3 + \left(17 a + 16\right)\cdot 41 + \left(3 a + 37\right)\cdot 41^{2} + \left(9 a + 4\right)\cdot 41^{3} + \left(4 a + 36\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 27 a + 30 + \left(38 a + 1\right)\cdot 41 + \left(36 a + 39\right)\cdot 41^{2} + \left(37 a + 24\right)\cdot 41^{3} + \left(37 a + 31\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 24 a + 13 + \left(23 a + 10\right)\cdot 41 + \left(37 a + 30\right)\cdot 41^{2} + \left(31 a + 28\right)\cdot 41^{3} + \left(36 a + 39\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 29 a + 22 + \left(a + 34\right)\cdot 41 + \left(32 a + 7\right)\cdot 41^{2} + \left(21 a + 26\right)\cdot 41^{3} + \left(19 a + 17\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.