Properties

Label 1.19.3t1.a.a
Dimension $1$
Group $C_3$
Conductor $19$
Root number not computed
Indicator $0$

Related objects

Learn more about

Basic invariants

Dimension: $1$
Group: $C_3$
Conductor: \(19\)
Artin number field: Galois closure of 3.3.361.1
Galois orbit size: $2$
Smallest permutation container: $C_3$
Parity: even
Dirichlet character: \(\chi_{19}(7,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{3} - x^{2} - 6 x + 7 $.

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 6\cdot 7 + 6\cdot 7^{2} + 6\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 + 6\cdot 7 + 5\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 5 + 7 + 6\cdot 7^{2} + 5\cdot 7^{3} + 2\cdot 7^{4} +O\left(7^{ 5 }\right)$

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.