Properties

Label 1.2359.6t1.a.b
Dimension $1$
Group $C_6$
Conductor $2359$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(2359\)\(\medspace = 7 \cdot 337 \)
Artin field: Galois closure of 6.6.91892879953.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{2359}(1684,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - 257x^{4} + 171x^{3} + 21257x^{2} - 7225x - 564733 \) 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:
$r_{ 1 }$ $=$ \( 10 a + 22 + \left(15 a + 9\right)\cdot 29 + \left(3 a + 12\right)\cdot 29^{2} + \left(12 a + 1\right)\cdot 29^{3} + \left(22 a + 24\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 19 a + \left(13 a + 22\right)\cdot 29 + 25 a\cdot 29^{2} + \left(16 a + 28\right)\cdot 29^{3} + \left(6 a + 2\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 10 a + 8 + \left(15 a + 13\right)\cdot 29 + \left(3 a + 27\right)\cdot 29^{2} + \left(12 a + 28\right)\cdot 29^{3} + \left(22 a + 18\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 10 a + 26 + \left(15 a + 21\right)\cdot 29 + \left(3 a + 14\right)\cdot 29^{2} + \left(12 a + 14\right)\cdot 29^{3} + \left(22 a + 24\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 19 a + 18 + \left(13 a + 1\right)\cdot 29 + \left(25 a + 17\right)\cdot 29^{2} + \left(16 a + 13\right)\cdot 29^{3} + \left(6 a + 8\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 19 a + 14 + \left(13 a + 18\right)\cdot 29 + \left(25 a + 14\right)\cdot 29^{2} + 16 a\cdot 29^{3} + \left(6 a + 8\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.