Properties

Label 1.35.6t1.a.b
Dimension $1$
Group $C_6$
Conductor $35$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(35\)\(\medspace = 5 \cdot 7 \)
Artin number field: Galois closure of 6.0.2100875.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{35}(24,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{6} - x^{5} + 8 x^{4} - 8 x^{3} + 22 x^{2} - 22 x + 29 $.

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $

Roots:
$r_{ 1 }$ $=$ $ 10 a + 19 + \left(33 a + 1\right)\cdot 41 + \left(27 a + 38\right)\cdot 41^{2} + \left(14 a + 39\right)\cdot 41^{3} + \left(9 a + 39\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 23 a + 29 + \left(40 a + 39\right)\cdot 41 + \left(22 a + 39\right)\cdot 41^{2} + \left(15 a + 2\right)\cdot 41^{3} + \left(8 a + 31\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 18 a + 16 + 15\cdot 41 + \left(18 a + 27\right)\cdot 41^{2} + \left(25 a + 26\right)\cdot 41^{3} + \left(32 a + 40\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 17 a + 21 + \left(27 a + 16\right)\cdot 41 + \left(34 a + 8\right)\cdot 41^{2} + \left(28 a + 34\right)\cdot 41^{3} + \left(7 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 24 a + 31 + \left(13 a + 40\right)\cdot 41 + \left(6 a + 2\right)\cdot 41^{2} + \left(12 a + 4\right)\cdot 41^{3} + \left(33 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 31 a + 8 + \left(7 a + 9\right)\cdot 41 + \left(13 a + 6\right)\cdot 41^{2} + \left(26 a + 15\right)\cdot 41^{3} + \left(31 a + 12\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.