# Properties

 Label 1.3e2_13.3t1.1c1 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 $f= x^{3} - 39 x - 91$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_3$ Parity: Even Corresponding Dirichlet character: $$\chi_{117}(61,\cdot)$$

## Galois action

### Roots of defining polynomial

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

### Generators of the action on the roots $r_{ 1 }, r_{ 2 }, r_{ 3 }$

 Cycle notation $(1,2,3)$

### Character values on conjugacy classes

 Size Order Action 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.