Properties

Label 1.161.6t1.b.b
Dimension $1$
Group $C_6$
Conductor $161$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(161\)\(\medspace = 7 \cdot 23 \)
Artin field: Galois closure of 6.0.29212967.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{161}(114,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} + 13x^{4} - 9x^{3} + 107x^{2} - 25x + 377 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 7 a + 5 + \left(30 a + 21\right)\cdot 43 + \left(10 a + 29\right)\cdot 43^{2} + \left(9 a + 23\right)\cdot 43^{3} + \left(4 a + 40\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 36 a + 23 + \left(12 a + 38\right)\cdot 43 + \left(32 a + 18\right)\cdot 43^{2} + \left(33 a + 24\right)\cdot 43^{3} + \left(38 a + 18\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 36 a + 19 + \left(12 a + 16\right)\cdot 43 + \left(32 a + 6\right)\cdot 43^{2} + \left(33 a + 37\right)\cdot 43^{3} + \left(38 a + 2\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 36 a + 12 + \left(12 a + 1\right)\cdot 43 + \left(32 a + 10\right)\cdot 43^{2} + \left(33 a + 22\right)\cdot 43^{3} + \left(38 a + 35\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 7 a + 16 + \left(30 a + 15\right)\cdot 43 + \left(10 a + 38\right)\cdot 43^{2} + \left(9 a + 25\right)\cdot 43^{3} + \left(4 a + 23\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 7 a + 12 + \left(30 a + 36\right)\cdot 43 + \left(10 a + 25\right)\cdot 43^{2} + \left(9 a + 38\right)\cdot 43^{3} + \left(4 a + 7\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.