Properties

Label 1.155.6t1.b.a
Dimension $1$
Group $C_6$
Conductor $155$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(155\)\(\medspace = 5 \cdot 31 \)
Artin field: 6.0.3578643875.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{155}(99,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} + 34 x^{4} - 73 x^{3} + 509 x^{2} - 470 x + 1396\)  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 $\mathbb{Q}_{ 29 }$: \(x^{2} + 24 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 a + 11 + \left(14 a + 13\right)\cdot 29 + \left(26 a + 16\right)\cdot 29^{2} + \left(15 a + 19\right)\cdot 29^{3} + \left(19 a + 5\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 14 + \left(a + 10\right)\cdot 29 + \left(26 a + 10\right)\cdot 29^{2} + \left(9 a + 27\right)\cdot 29^{3} + \left(28 a + 2\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 26 a + 11 + 12\cdot 29 + \left(15 a + 15\right)\cdot 29^{2} + \left(a + 5\right)\cdot 29^{3} + \left(8 a + 10\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 25 + \left(28 a + 19\right)\cdot 29 + \left(13 a + 2\right)\cdot 29^{2} + \left(27 a + 27\right)\cdot 29^{3} + \left(20 a + 19\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 24 a + 7 + \left(14 a + 21\right)\cdot 29 + \left(2 a + 18\right)\cdot 29^{2} + \left(13 a + 14\right)\cdot 29^{3} + 9 a\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 22 a + 20 + \left(27 a + 9\right)\cdot 29 + \left(2 a + 23\right)\cdot 29^{2} + \left(19 a + 21\right)\cdot 29^{3} + 18\cdot 29^{4} +O(29^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.