Properties

Label 1.171.3t1.a
Dimension $1$
Group $C_3$
Conductor $171$
Indicator $0$

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Basic invariants

Dimension:$1$
Group:$C_3$
Conductor:\(171\)\(\medspace = 3^{2} \cdot 19 \)
Artin number field: Galois closure of 3.3.29241.1
Galois orbit size: $2$
Smallest permutation container: $C_3$
Parity: even
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 1 + 5\cdot 13 + 7\cdot 13^{2} + 12\cdot 13^{3} + 9\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 10\cdot 13 + 7\cdot 13^{2} + 10\cdot 13^{3} +O(13^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 7 + 10\cdot 13 + 10\cdot 13^{2} + 2\cdot 13^{3} + 2\cdot 13^{4} +O(13^{5})\) 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 values
$c1$ $c2$
$1$ $1$ $()$ $1$ $1$
$1$ $3$ $(1,2,3)$ $\zeta_{3}$ $-\zeta_{3} - 1$
$1$ $3$ $(1,3,2)$ $-\zeta_{3} - 1$ $\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.