# Properties

 Label 2.3_2351.3t2.1c1 Dimension 2 Group $S_3$ Conductor $3 \cdot 2351$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $7053= 3 \cdot 2351$ Artin number field: Splitting field of $f= x^{3} - x^{2} - 23 x + 48$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_3$ Parity: Even Determinant: 1.3_2351.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $19 + 36\cdot 41 + 29\cdot 41^{2} + 30\cdot 41^{3} + 14\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 2 }$ $=$ $25 + 33\cdot 41 + 2\cdot 41^{2} + 35\cdot 41^{3} + 37\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 3 }$ $=$ $39 + 11\cdot 41 + 8\cdot 41^{2} + 16\cdot 41^{3} + 29\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

 Cycle notation $(1,2,3)$ $(1,2)$

### Character values on conjugacy classes

 Size Order Action on $r_{ 1 }, r_{ 2 }, r_{ 3 }$ Character value $1$ $1$ $()$ $2$ $3$ $2$ $(1,2)$ $0$ $2$ $3$ $(1,2,3)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.