# Properties

 Label 2.140.3t2.a Dimension $2$ Group $S_3$ Conductor $140$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$140$$$$\medspace = 2^{2} \cdot 5 \cdot 7$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 3.1.140.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.140.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$7 + 50\cdot 71 + 11\cdot 71^{2} + 54\cdot 71^{3} + 50\cdot 71^{4} +O(71^{5})$$ 7 + 50*71 + 11*71^2 + 54*71^3 + 50*71^4+O(71^5) $r_{ 2 }$ $=$ $$21 + 38\cdot 71 + 66\cdot 71^{2} + 42\cdot 71^{3} + 49\cdot 71^{4} +O(71^{5})$$ 21 + 38*71 + 66*71^2 + 42*71^3 + 49*71^4+O(71^5) $r_{ 3 }$ $=$ $$43 + 53\cdot 71 + 63\cdot 71^{2} + 44\cdot 71^{3} + 41\cdot 71^{4} +O(71^{5})$$ 43 + 53*71 + 63*71^2 + 44*71^3 + 41*71^4+O(71^5)

### 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 values $c1$ $1$ $1$ $()$ $2$ $3$ $2$ $(1,2)$ $0$ $2$ $3$ $(1,2,3)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.