Properties

Label 1.3e2_13.3t1.2c2
Dimension 1
Group $C_3$
Conductor $ 3^{2} \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_3$
Conductor:$117= 3^{2} \cdot 13 $
Artin number field: Splitting field of $f= x^{3} - 39 x - 26 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_3$
Parity: Even
Corresponding Dirichlet character: \(\chi_{117}(22,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 6 + 5\cdot 11 + 11^{2} + 10\cdot 11^{3} + 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 + 2\cdot 11 + 8\cdot 11^{2} + 10\cdot 11^{3} + 3\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 9 + 2\cdot 11 + 11^{2} + 11^{3} + 5\cdot 11^{4} +O\left(11^{ 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.