Properties

Label 1.7.3t1.a.a
Dimension $1$
Group $C_3$
Conductor $7$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_3$
Conductor: \(7\)
Artin field: \(\Q(\zeta_{7})^+\)
Galois orbit size: $2$
Smallest permutation container: $C_3$
Parity: even
Dirichlet character: \(\chi_{7}(2,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{3} - x^{2} - 2 x + 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 3 + 2\cdot 13 + 5\cdot 13^{2} + 5\cdot 13^{3} + 11\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 10\cdot 13 + 3\cdot 13^{2} + 8\cdot 13^{3} + 12\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 6 + 4\cdot 13^{2} + 12\cdot 13^{3} + 13^{4} +O(13^{5})\)  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$$3$$(1,3,2)$$-\zeta_{3} - 1$

The blue line marks the conjugacy class containing complex conjugation.