Properties

Label 1.203.6t1.b.b
Dimension $1$
Group $C_6$
Conductor $203$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(203\)\(\medspace = 7 \cdot 29 \)
Artin field: Galois closure of 6.6.58557989.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{203}(86,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - 26x^{4} + 17x^{3} + 159x^{2} - 64x - 169 \) Copy content Toggle raw display .

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 38 a + 19 + \left(37 a + 33\right)\cdot 41 + \left(19 a + 26\right)\cdot 41^{2} + \left(23 a + 1\right)\cdot 41^{3} + \left(26 a + 22\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 a + 33 + \left(3 a + 23\right)\cdot 41 + \left(21 a + 25\right)\cdot 41^{2} + \left(17 a + 36\right)\cdot 41^{3} + \left(14 a + 17\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 38 a + 35 + \left(37 a + 27\right)\cdot 41 + \left(19 a + 18\right)\cdot 41^{2} + \left(23 a + 18\right)\cdot 41^{3} + \left(26 a + 34\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 26 + \left(3 a + 21\right)\cdot 41 + \left(21 a + 40\right)\cdot 41^{2} + \left(17 a + 27\right)\cdot 41^{3} + \left(14 a + 8\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a + 10 + \left(3 a + 27\right)\cdot 41 + \left(21 a + 7\right)\cdot 41^{2} + \left(17 a + 11\right)\cdot 41^{3} + \left(14 a + 37\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 38 a + 1 + \left(37 a + 30\right)\cdot 41 + \left(19 a + 3\right)\cdot 41^{2} + \left(23 a + 27\right)\cdot 41^{3} + \left(26 a + 2\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.