# Properties

 Label 2.455.6t5.a.a Dimension $2$ Group $S_3\times C_3$ Conductor $455$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$455$$$$\medspace = 5 \cdot 7 \cdot 13$$ Artin stem field: 6.0.7245875.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.455.6t1.b.a Projective image: $S_3$ Projective stem field: 3.1.5915.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + 2 x^{4} - 13 x^{3} + 3 x^{2} + 14 x + 9$$  .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.

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

Roots:
 $r_{ 1 }$ $=$ $$a + \left(17 a + 7\right)\cdot 31 + \left(12 a + 21\right)\cdot 31^{2} + \left(17 a + 6\right)\cdot 31^{3} + \left(7 a + 16\right)\cdot 31^{4} + \left(25 a + 30\right)\cdot 31^{5} +O(31^{6})$$ $r_{ 2 }$ $=$ $$30 a + 2 + \left(13 a + 9\right)\cdot 31 + \left(18 a + 29\right)\cdot 31^{2} + \left(13 a + 28\right)\cdot 31^{3} + \left(23 a + 13\right)\cdot 31^{4} + \left(5 a + 11\right)\cdot 31^{5} +O(31^{6})$$ $r_{ 3 }$ $=$ $$7 a + 27 + \left(20 a + 26\right)\cdot 31 + \left(14 a + 25\right)\cdot 31^{2} + \left(15 a + 15\right)\cdot 31^{3} + \left(14 a + 3\right)\cdot 31^{4} + \left(5 a + 19\right)\cdot 31^{5} +O(31^{6})$$ $r_{ 4 }$ $=$ $$24 a + 10 + \left(10 a + 29\right)\cdot 31 + \left(16 a + 3\right)\cdot 31^{2} + \left(15 a + 1\right)\cdot 31^{3} + \left(16 a + 17\right)\cdot 31^{4} + \left(25 a + 15\right)\cdot 31^{5} +O(31^{6})$$ $r_{ 5 }$ $=$ $$12 a + \left(19 a + 28\right)\cdot 31 + \left(20 a + 10\right)\cdot 31^{2} + \left(22 a + 23\right)\cdot 31^{3} + \left(16 a + 15\right)\cdot 31^{4} + \left(10 a + 21\right)\cdot 31^{5} +O(31^{6})$$ $r_{ 6 }$ $=$ $$19 a + 24 + \left(11 a + 23\right)\cdot 31 + \left(10 a + 1\right)\cdot 31^{2} + \left(8 a + 17\right)\cdot 31^{3} + \left(14 a + 26\right)\cdot 31^{4} + \left(20 a + 25\right)\cdot 31^{5} +O(31^{6})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.