# Properties

 Label 1.13.6t1.a.a Dimension $1$ Group $C_6$ Conductor $13$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$13$$ Artin field: Galois closure of $$\Q(\zeta_{13})^+$$ Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Dirichlet character: $$\chi_{13}(10,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} - 5x^{4} + 4x^{3} + 6x^{2} - 3x - 1$$ x^6 - x^5 - 5*x^4 + 4*x^3 + 6*x^2 - 3*x - 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:
 $r_{ 1 }$ $=$ $$3 a + 4\cdot 31 + \left(11 a + 23\right)\cdot 31^{2} + 4\cdot 31^{3} + \left(4 a + 10\right)\cdot 31^{4} +O(31^{5})$$ 3*a + 4*31 + (11*a + 23)*31^2 + 4*31^3 + (4*a + 10)*31^4+O(31^5) $r_{ 2 }$ $=$ $$28 a + 6 + \left(30 a + 1\right)\cdot 31 + \left(19 a + 14\right)\cdot 31^{2} + \left(30 a + 25\right)\cdot 31^{3} + \left(26 a + 17\right)\cdot 31^{4} +O(31^{5})$$ 28*a + 6 + (30*a + 1)*31 + (19*a + 14)*31^2 + (30*a + 25)*31^3 + (26*a + 17)*31^4+O(31^5) $r_{ 3 }$ $=$ $$24 a + 9 + \left(2 a + 12\right)\cdot 31 + \left(14 a + 18\right)\cdot 31^{2} + \left(27 a + 15\right)\cdot 31^{3} + 15 a\cdot 31^{4} +O(31^{5})$$ 24*a + 9 + (2*a + 12)*31 + (14*a + 18)*31^2 + (27*a + 15)*31^3 + 15*a*31^4+O(31^5) $r_{ 4 }$ $=$ $$18 a + 24 + \left(15 a + 18\right)\cdot 31 + \left(22 a + 28\right)\cdot 31^{2} + \left(23 a + 13\right)\cdot 31^{3} + \left(20 a + 5\right)\cdot 31^{4} +O(31^{5})$$ 18*a + 24 + (15*a + 18)*31 + (22*a + 28)*31^2 + (23*a + 13)*31^3 + (20*a + 5)*31^4+O(31^5) $r_{ 5 }$ $=$ $$13 a + 29 + 15 a\cdot 31 + \left(8 a + 27\right)\cdot 31^{2} + \left(7 a + 7\right)\cdot 31^{3} + \left(10 a + 23\right)\cdot 31^{4} +O(31^{5})$$ 13*a + 29 + 15*a*31 + (8*a + 27)*31^2 + (7*a + 7)*31^3 + (10*a + 23)*31^4+O(31^5) $r_{ 6 }$ $=$ $$7 a + 26 + \left(28 a + 24\right)\cdot 31 + \left(16 a + 12\right)\cdot 31^{2} + \left(3 a + 25\right)\cdot 31^{3} + \left(15 a + 4\right)\cdot 31^{4} +O(31^{5})$$ 7*a + 26 + (28*a + 24)*31 + (16*a + 12)*31^2 + (3*a + 25)*31^3 + (15*a + 4)*31^4+O(31^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.