Properties

Label 1.13.3t1.1c2
Dimension 1
Group $C_3$
Conductor $ 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_3$
Conductor:$13 $
Artin number field: Splitting field of $f= x^{3} - x^{2} - 4 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_3$
Parity: Even
Corresponding Dirichlet character: \(\chi_{13}(9,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 1 + 3\cdot 5 + 2\cdot 5^{2} + 5^{3} + 3\cdot 5^{4} +O\left(5^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 + 4\cdot 5 + 3\cdot 5^{2} + 3\cdot 5^{3} + 2\cdot 5^{4} +O\left(5^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 3 + 2\cdot 5 + 3\cdot 5^{2} + 4\cdot 5^{3} + 3\cdot 5^{4} +O\left(5^{ 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$
$1$$3$$(1,3,2)$$\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.