Properties

Label 1.140.6t1.b.a
Dimension $1$
Group $C_6$
Conductor $140$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \)
Artin field: 6.0.19208000.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{140}(39,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} + 12 x^{4} - 14 x^{3} + 87 x^{2} - 64 x + 281\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{2} + 12 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a + 11 + \left(12 a + 4\right)\cdot 13 + \left(11 a + 10\right)\cdot 13^{2} + \left(12 a + 7\right)\cdot 13^{3} + \left(12 a + 12\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 12 a + 12 + 2\cdot 13 + \left(a + 10\right)\cdot 13^{2} + 8\cdot 13^{3} + 12\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( a + 9 + \left(12 a + 9\right)\cdot 13 + \left(11 a + 11\right)\cdot 13^{2} + \left(12 a + 4\right)\cdot 13^{3} + \left(12 a + 11\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 12 a + 6\cdot 13 + \left(a + 10\right)\cdot 13^{2} + 12\cdot 13^{3} + 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( a + 12 + \left(12 a + 7\right)\cdot 13 + \left(11 a + 10\right)\cdot 13^{2} + \left(12 a + 11\right)\cdot 13^{3} + \left(12 a + 1\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 12 a + 10 + 7\cdot 13 + \left(a + 11\right)\cdot 13^{2} + 5\cdot 13^{3} + 11\cdot 13^{4} +O(13^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.