Properties

Label 1.91.6t1.j.a
Dimension $1$
Group $C_6$
Conductor $91$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(91\)\(\medspace = 7 \cdot 13 \)
Artin field: 6.6.5274997.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{91}(25,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} - 14 x^{4} + 9 x^{3} + 35 x^{2} - 16 x - 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \(x^{2} + 38 x + 6\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 12 a + 40 + \left(29 a + 29\right)\cdot 41 + \left(40 a + 29\right)\cdot 41^{2} + 9\cdot 41^{3} + \left(37 a + 37\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 12 a + 17 + \left(29 a + 33\right)\cdot 41 + \left(40 a + 11\right)\cdot 41^{2} + 25\cdot 41^{3} + \left(37 a + 15\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 29 a + 28 + \left(11 a + 21\right)\cdot 41 + 14\cdot 41^{2} + \left(40 a + 4\right)\cdot 41^{3} + \left(3 a + 15\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 29 a + 12 + \left(11 a + 27\right)\cdot 41 + 22\cdot 41^{2} + \left(40 a + 28\right)\cdot 41^{3} + \left(3 a + 2\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 29 a + 35 + \left(11 a + 23\right)\cdot 41 + 40\cdot 41^{2} + \left(40 a + 12\right)\cdot 41^{3} + \left(3 a + 24\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 12 a + 33 + \left(29 a + 27\right)\cdot 41 + \left(40 a + 3\right)\cdot 41^{2} + 41^{3} + \left(37 a + 28\right)\cdot 41^{4} +O(41^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.