Properties

Label 1.760.6t1.a.b
Dimension $1$
Group $C_6$
Conductor $760$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(760\)\(\medspace = 2^{3} \cdot 5 \cdot 19 \)
Artin field: 6.6.8340544000.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{760}(349,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} - 41 x^{4} + 66 x^{3} + 302 x^{2} + 16 x - 151\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \(x^{2} + 6 x + 3\)  Toggle raw display

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

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

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

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$$3$$(1,5,6)(2,3,4)$$-\zeta_{3} - 1$
$1$$3$$(1,6,5)(2,4,3)$$\zeta_{3}$
$1$$6$$(1,3,5,4,6,2)$$-\zeta_{3}$
$1$$6$$(1,2,6,4,5,3)$$\zeta_{3} + 1$

The blue line marks the conjugacy class containing complex conjugation.