Properties

Label 1.313.3t1.a.b
Dimension $1$
Group $C_3$
Conductor $313$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_3$
Conductor: \(313\)
Artin field: Galois closure of 3.3.97969.1
Galois orbit size: $2$
Smallest permutation container: $C_3$
Parity: even
Dirichlet character: \(\chi_{313}(214,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{3} - x^{2} - 104x - 371 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 5 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 1 + 2\cdot 5^{2} + 3\cdot 5^{4} + 5^{5} +O(5^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 2 + 2\cdot 5^{2} + 2\cdot 5^{3} + 3\cdot 5^{4} + 2\cdot 5^{5} +O(5^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 3 + 4\cdot 5 + 2\cdot 5^{3} + 3\cdot 5^{4} +O(5^{6})\) 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.