Properties

Label 1.65.6t1.a.b
Dimension $1$
Group $C_6$
Conductor $65$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(65\)\(\medspace = 5 \cdot 13 \)
Artin field: Galois closure of 6.6.46411625.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{65}(49,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - 18x^{4} + 17x^{3} + 58x^{2} - 16x - 1 \) 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:
$r_{ 1 }$ $=$ \( 20 a + 13 + \left(14 a + 29\right)\cdot 31 + \left(30 a + 7\right)\cdot 31^{2} + \left(7 a + 12\right)\cdot 31^{3} + 6 a\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a + 27 + \left(14 a + 2\right)\cdot 31 + \left(27 a + 23\right)\cdot 31^{2} + \left(2 a + 21\right)\cdot 31^{3} + \left(25 a + 21\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 11 a + 22 + \left(16 a + 7\right)\cdot 31 + 23\cdot 31^{2} + \left(23 a + 28\right)\cdot 31^{3} + \left(24 a + 4\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 16 a + 26 + \left(16 a + 16\right)\cdot 31 + \left(3 a + 1\right)\cdot 31^{2} + 28 a\cdot 31^{3} + \left(5 a + 7\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 18 a + 16 + \left(23 a + 3\right)\cdot 31 + \left(23 a + 22\right)\cdot 31^{2} + \left(11 a + 30\right)\cdot 31^{3} + \left(9 a + 25\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 13 a + 21 + \left(7 a + 1\right)\cdot 31 + \left(7 a + 15\right)\cdot 31^{2} + \left(19 a + 30\right)\cdot 31^{3} + \left(21 a + 1\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.