Properties

Label 1.280.6t1.c.b
Dimension $1$
Group $C_6$
Conductor $280$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \)
Artin field: 6.0.1075648000.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{280}(229,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} + 70 x^{4} + 1400 x^{2} + 7000\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.

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

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

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

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

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

The blue line marks the conjugacy class containing complex conjugation.