Properties

Label 1.260.6t1.a.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 6.0.228488000.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{260}(159,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + 8x^{4} - 14x^{3} + 103x^{2} + 28x + 421 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 24 a + 11 + \left(23 a + 25\right)\cdot 31 + \left(10 a + 16\right)\cdot 31^{2} + \left(5 a + 25\right)\cdot 31^{3} + \left(2 a + 5\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 15 + 7 a\cdot 31 + \left(20 a + 8\right)\cdot 31^{2} + \left(25 a + 6\right)\cdot 31^{3} + \left(28 a + 28\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 7 a + 28 + \left(7 a + 17\right)\cdot 31 + \left(20 a + 14\right)\cdot 31^{2} + \left(25 a + 25\right)\cdot 31^{3} + \left(28 a + 4\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 a + 30 + \left(7 a + 16\right)\cdot 31 + \left(20 a + 20\right)\cdot 31^{2} + \left(25 a + 14\right)\cdot 31^{3} + \left(28 a + 27\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 24 a + 13 + \left(23 a + 24\right)\cdot 31 + \left(10 a + 22\right)\cdot 31^{2} + \left(5 a + 14\right)\cdot 31^{3} + \left(2 a + 28\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 24 a + 29 + \left(23 a + 7\right)\cdot 31 + \left(10 a + 10\right)\cdot 31^{2} + \left(5 a + 6\right)\cdot 31^{3} + \left(2 a + 29\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.