# Properties

 Label 1.52.6t1.b.a Dimension $1$ Group $C_6$ Conductor $52$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$52$$$$\medspace = 2^{2} \cdot 13$$ Artin field: Galois closure of 6.0.1827904.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{52}(3,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} + 9x^{4} + 14x^{2} + 1$$ x^6 + 9*x^4 + 14*x^2 + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $$x^{2} + 29x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$24 a + 7 + \left(2 a + 9\right)\cdot 31 + \left(17 a + 15\right)\cdot 31^{2} + \left(30 a + 24\right)\cdot 31^{3} + \left(3 a + 26\right)\cdot 31^{4} +O(31^{5})$$ 24*a + 7 + (2*a + 9)*31 + (17*a + 15)*31^2 + (30*a + 24)*31^3 + (3*a + 26)*31^4+O(31^5) $r_{ 2 }$ $=$ $$5 a + 26 + \left(18 a + 30\right)\cdot 31 + \left(3 a + 20\right)\cdot 31^{2} + \left(18 a + 14\right)\cdot 31^{3} + 8\cdot 31^{4} +O(31^{5})$$ 5*a + 26 + (18*a + 30)*31 + (3*a + 20)*31^2 + (18*a + 14)*31^3 + 8*31^4+O(31^5) $r_{ 3 }$ $=$ $$15 a + 16 + \left(18 a + 4\right)\cdot 31 + \left(22 a + 2\right)\cdot 31^{2} + \left(11 a + 15\right)\cdot 31^{3} + \left(13 a + 23\right)\cdot 31^{4} +O(31^{5})$$ 15*a + 16 + (18*a + 4)*31 + (22*a + 2)*31^2 + (11*a + 15)*31^3 + (13*a + 23)*31^4+O(31^5) $r_{ 4 }$ $=$ $$7 a + 24 + \left(28 a + 21\right)\cdot 31 + \left(13 a + 15\right)\cdot 31^{2} + 6\cdot 31^{3} + \left(27 a + 4\right)\cdot 31^{4} +O(31^{5})$$ 7*a + 24 + (28*a + 21)*31 + (13*a + 15)*31^2 + 6*31^3 + (27*a + 4)*31^4+O(31^5) $r_{ 5 }$ $=$ $$26 a + 5 + 12 a\cdot 31 + \left(27 a + 10\right)\cdot 31^{2} + \left(12 a + 16\right)\cdot 31^{3} + \left(30 a + 22\right)\cdot 31^{4} +O(31^{5})$$ 26*a + 5 + 12*a*31 + (27*a + 10)*31^2 + (12*a + 16)*31^3 + (30*a + 22)*31^4+O(31^5) $r_{ 6 }$ $=$ $$16 a + 15 + \left(12 a + 26\right)\cdot 31 + \left(8 a + 28\right)\cdot 31^{2} + \left(19 a + 15\right)\cdot 31^{3} + \left(17 a + 7\right)\cdot 31^{4} +O(31^{5})$$ 16*a + 15 + (12*a + 26)*31 + (8*a + 28)*31^2 + (19*a + 15)*31^3 + (17*a + 7)*31^4+O(31^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.