Properties

Label 1.260.6t1.b.a
Dimension $1$
Group $C_6$
Conductor $260$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(260\)\(\medspace = 2^{2} \cdot 5 \cdot 13 \)
Artin field: Galois closure of 6.0.2970344000.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{260}(179,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} + 65x^{4} + 650x^{2} + 1625 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 14 a + 33 + \left(5 a + 1\right)\cdot 47 + \left(17 a + 9\right)\cdot 47^{2} + \left(20 a + 35\right)\cdot 47^{3} + \left(23 a + 33\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 a + 39 + \left(14 a + 36\right)\cdot 47 + \left(36 a + 17\right)\cdot 47^{2} + \left(37 a + 27\right)\cdot 47^{3} + \left(25 a + 16\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 12 a + 35 + \left(17 a + 35\right)\cdot 47 + \left(28 a + 3\right)\cdot 47^{2} + \left(44 a + 40\right)\cdot 47^{3} + \left(29 a + 15\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 33 a + 14 + \left(41 a + 45\right)\cdot 47 + \left(29 a + 37\right)\cdot 47^{2} + \left(26 a + 11\right)\cdot 47^{3} + \left(23 a + 13\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 39 a + 8 + \left(32 a + 10\right)\cdot 47 + \left(10 a + 29\right)\cdot 47^{2} + \left(9 a + 19\right)\cdot 47^{3} + \left(21 a + 30\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 35 a + 12 + \left(29 a + 11\right)\cdot 47 + \left(18 a + 43\right)\cdot 47^{2} + \left(2 a + 6\right)\cdot 47^{3} + \left(17 a + 31\right)\cdot 47^{4} +O(47^{5})\) 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)$$\zeta_{3}$
$1$$3$$(1,3,2)(4,6,5)$$-\zeta_{3} - 1$
$1$$6$$(1,5,3,4,2,6)$$-\zeta_{3}$
$1$$6$$(1,6,2,4,3,5)$$\zeta_{3} + 1$

The blue line marks the conjugacy class containing complex conjugation.