Properties

Label 1.104.6t1.a.a
Dimension $1$
Group $C_6$
Conductor $104$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(104\)\(\medspace = 2^{3} \cdot 13 \)
Artin field: 6.6.14623232.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{104}(29,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} - 13 x^{4} + 14 x^{3} + 26 x^{2} - 28 x + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \(x^{2} + 49 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 52 a + 16 + \left(25 a + 15\right)\cdot 53 + \left(29 a + 37\right)\cdot 53^{2} + \left(34 a + 49\right)\cdot 53^{3} + \left(38 a + 19\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 52 a + 35 + \left(25 a + 29\right)\cdot 53 + \left(29 a + 52\right)\cdot 53^{2} + \left(34 a + 37\right)\cdot 53^{3} + \left(38 a + 50\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( a + 5 + \left(27 a + 35\right)\cdot 53 + \left(23 a + 49\right)\cdot 53^{2} + \left(18 a + 16\right)\cdot 53^{3} + \left(14 a + 28\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 52 a + 9 + \left(25 a + 36\right)\cdot 53 + \left(29 a + 10\right)\cdot 53^{2} + \left(34 a + 14\right)\cdot 53^{3} + \left(38 a + 14\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( a + 31 + \left(27 a + 28\right)\cdot 53 + \left(23 a + 38\right)\cdot 53^{2} + \left(18 a + 40\right)\cdot 53^{3} + \left(14 a + 11\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( a + 12 + \left(27 a + 14\right)\cdot 53 + \left(23 a + 23\right)\cdot 53^{2} + \left(18 a + 52\right)\cdot 53^{3} + \left(14 a + 33\right)\cdot 53^{4} +O(53^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.