Properties

Label 1.57.6t1.a.b
Dimension $1$
Group $C_6$
Conductor $57$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(57\)\(\medspace = 3 \cdot 19 \)
Artin number field: Galois closure of 6.0.3518667.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{57}(11,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{6} - x^{5} + 7 x^{4} - 8 x^{3} + 43 x^{2} - 42 x + 49 $.

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $ x^{2} + 7 x + 2 $

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.