Properties

 Label 1.10039.3t1.1c2 Dimension 1 Group $C_3$ Conductor $10039$ Root number not computed Frobenius-Schur indicator 0

Related objects

Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $10039$ Artin number field: Splitting field of $f= x^{3} - x^{2} - 3346 x + 56144$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_3$ Parity: Even Corresponding Dirichlet character: $$\chi_{10039}(6307,\cdot)$$

Galois action

Roots of defining polynomial

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

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 $1$ $1$ $()$ $1$ $1$ $3$ $(1,2,3)$ $-\zeta_{3} - 1$ $1$ $3$ $(1,3,2)$ $\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.