Properties

Label 1.1757.6t1.b.a
Dimension $1$
Group $C_6$
Conductor $1757$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(1757\)\(\medspace = 7 \cdot 251 \)
Artin field: Galois closure of 6.0.37967615651.3
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{1757}(501,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} + 184x^{4} - 123x^{3} + 11849x^{2} - 3844x + 266111 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 a + 28 + 3\cdot 29 + \left(20 a + 22\right)\cdot 29^{2} + 26 a\cdot 29^{3} + \left(7 a + 19\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 2 a + 17 + 12\cdot 29 + \left(20 a + 9\right)\cdot 29^{2} + \left(26 a + 15\right)\cdot 29^{3} + \left(7 a + 24\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 13 + \left(20 a + 7\right)\cdot 29^{2} + \left(26 a + 2\right)\cdot 29^{3} + \left(7 a + 24\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 27 a + 9 + \left(28 a + 2\right)\cdot 29 + \left(8 a + 6\right)\cdot 29^{2} + \left(2 a + 27\right)\cdot 29^{3} + \left(21 a + 2\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 27 a + 23 + \left(28 a + 27\right)\cdot 29 + \left(8 a + 19\right)\cdot 29^{2} + \left(2 a + 28\right)\cdot 29^{3} + \left(21 a + 7\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 27 a + 27 + \left(28 a + 10\right)\cdot 29 + \left(8 a + 22\right)\cdot 29^{2} + \left(2 a + 12\right)\cdot 29^{3} + \left(21 a + 8\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.