Properties

Label 1.119.6t1.b.b
Dimension $1$
Group $C_6$
Conductor $119$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(119\)\(\medspace = 7 \cdot 17 \)
Artin field: Galois closure of 6.6.11796113.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{119}(67,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - 17x^{4} + 11x^{3} + 57x^{2} - 25x - 13 \) Copy content 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 $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a + 5 + \left(2 a + 7\right)\cdot 29 + \left(17 a + 3\right)\cdot 29^{2} + \left(8 a + 1\right)\cdot 29^{3} + \left(7 a + 17\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( a + 1 + \left(2 a + 24\right)\cdot 29 + 17 a\cdot 29^{2} + \left(8 a + 17\right)\cdot 29^{3} + \left(7 a + 16\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( a + 16 + \left(2 a + 27\right)\cdot 29 + \left(17 a + 15\right)\cdot 29^{2} + \left(8 a + 15\right)\cdot 29^{3} + \left(7 a + 11\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 28 a + 10 + \left(26 a + 16\right)\cdot 29 + \left(11 a + 28\right)\cdot 29^{2} + \left(20 a + 26\right)\cdot 29^{3} + \left(21 a + 15\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 28 a + 6 + \left(26 a + 4\right)\cdot 29 + \left(11 a + 26\right)\cdot 29^{2} + \left(20 a + 13\right)\cdot 29^{3} + \left(21 a + 15\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 28 a + 21 + \left(26 a + 7\right)\cdot 29 + \left(11 a + 12\right)\cdot 29^{2} + \left(20 a + 12\right)\cdot 29^{3} + \left(21 a + 10\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.