Properties

Label 1.171.3t1.b.b
Dimension $1$
Group $C_3$
Conductor $171$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_3$
Conductor: \(171\)\(\medspace = 3^{2} \cdot 19 \)
Artin field: Galois closure of 3.3.29241.2
Galois orbit size: $2$
Smallest permutation container: $C_3$
Parity: even
Dirichlet character: \(\chi_{171}(106,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{3} - 57x - 19 \) Copy content Toggle raw display .

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

Roots:
$r_{ 1 }$ $=$ \( 6 + 20\cdot 29 + 2\cdot 29^{2} + 29^{3} + 11\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 25 + 6\cdot 29 + 15\cdot 29^{2} + 26\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 27 + 29 + 11\cdot 29^{2} + 29^{3} + 6\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display

Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $

Cycle notation
$(1,2,3)$

Character values on conjugacy classes

SizeOrderAction on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ Character value
$1$$1$$()$$1$
$1$$3$$(1,2,3)$$-\zeta_{3} - 1$
$1$$3$$(1,3,2)$$\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.