# Properties

 Label 2.135.3t2.a.a Dimension $2$ Group $S_3$ Conductor $135$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$135$$$$\medspace = 3^{3} \cdot 5$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin field: Galois closure of 6.0.273375.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Determinant: 1.15.2t1.a.a Projective image: $S_3$ Projective field: Galois closure of 6.0.273375.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 3x^{5} + 3x^{4} - x^{3} + 3x^{2} - 3x + 1$$ x^6 - 3*x^5 + 3*x^4 - x^3 + 3*x^2 - 3*x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $$x^{2} + 6x + 3$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.