# Properties

 Label 2.104.3t2.a.a Dimension $2$ Group $S_3$ Conductor $104$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$104$$$$\medspace = 2^{3} \cdot 13$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin field: Galois closure of 6.0.1124864.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Determinant: 1.104.2t1.b.a Projective image: $S_3$ Projective field: Galois closure of 6.0.1124864.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} + 8x^{4} + 29x^{2} + 26$$ x^6 + 8*x^4 + 29*x^2 + 26 .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$7 a + 4 + \left(a + 8\right)\cdot 11 + \left(6 a + 1\right)\cdot 11^{2} + \left(2 a + 5\right)\cdot 11^{3} + \left(2 a + 5\right)\cdot 11^{4} +O(11^{5})$$ 7*a + 4 + (a + 8)*11 + (6*a + 1)*11^2 + (2*a + 5)*11^3 + (2*a + 5)*11^4+O(11^5) $r_{ 2 }$ $=$ $$7 a + 1 + \left(a + 3\right)\cdot 11 + \left(6 a + 8\right)\cdot 11^{2} + \left(2 a + 1\right)\cdot 11^{3} + \left(2 a + 10\right)\cdot 11^{4} +O(11^{5})$$ 7*a + 1 + (a + 3)*11 + (6*a + 8)*11^2 + (2*a + 1)*11^3 + (2*a + 10)*11^4+O(11^5) $r_{ 3 }$ $=$ $$6 a + 10 + 6 a\cdot 11 + a\cdot 11^{2} + \left(6 a + 5\right)\cdot 11^{3} + \left(2 a + 3\right)\cdot 11^{4} +O(11^{5})$$ 6*a + 10 + 6*a*11 + a*11^2 + (6*a + 5)*11^3 + (2*a + 3)*11^4+O(11^5) $r_{ 4 }$ $=$ $$4 a + 7 + \left(9 a + 2\right)\cdot 11 + \left(4 a + 9\right)\cdot 11^{2} + \left(8 a + 5\right)\cdot 11^{3} + \left(8 a + 5\right)\cdot 11^{4} +O(11^{5})$$ 4*a + 7 + (9*a + 2)*11 + (4*a + 9)*11^2 + (8*a + 5)*11^3 + (8*a + 5)*11^4+O(11^5) $r_{ 5 }$ $=$ $$4 a + 10 + \left(9 a + 7\right)\cdot 11 + \left(4 a + 2\right)\cdot 11^{2} + \left(8 a + 9\right)\cdot 11^{3} + 8 a\cdot 11^{4} +O(11^{5})$$ 4*a + 10 + (9*a + 7)*11 + (4*a + 2)*11^2 + (8*a + 9)*11^3 + 8*a*11^4+O(11^5) $r_{ 6 }$ $=$ $$5 a + 1 + \left(4 a + 10\right)\cdot 11 + \left(9 a + 10\right)\cdot 11^{2} + \left(4 a + 5\right)\cdot 11^{3} + \left(8 a + 7\right)\cdot 11^{4} +O(11^{5})$$ 5*a + 1 + (4*a + 10)*11 + (9*a + 10)*11^2 + (4*a + 5)*11^3 + (8*a + 7)*11^4+O(11^5)

## Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,2,6)(3,4,5)$ $(1,3)(2,5)(4,6)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $2$ $3$ $2$ $(1,3)(2,5)(4,6)$ $0$ $2$ $3$ $(1,2,6)(3,4,5)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.