# Properties

 Label 2.1148.6t5.b.b Dimension $2$ Group $S_3\times C_3$ Conductor $1148$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$1148$$$$\medspace = 2^{2} \cdot 7 \cdot 41$$ Artin stem field: 6.0.216136256.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.1148.6t1.a.b Projective image: $S_3$ Projective stem field: 3.1.8036.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2 x^{5} - 3 x^{4} + 8 x^{3} - 8 x + 45$$  .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{2} + 12 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$a + \left(9 a + 6\right)\cdot 13 + \left(a + 11\right)\cdot 13^{2} + \left(10 a + 6\right)\cdot 13^{3} + \left(7 a + 12\right)\cdot 13^{4} + \left(4 a + 8\right)\cdot 13^{5} + \left(10 a + 12\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 2 }$ $=$ $$12 a + 1 + \left(3 a + 1\right)\cdot 13 + \left(11 a + 4\right)\cdot 13^{2} + \left(2 a + 2\right)\cdot 13^{3} + \left(5 a + 10\right)\cdot 13^{4} + \left(8 a + 5\right)\cdot 13^{5} + \left(2 a + 5\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 3 }$ $=$ $$11 a + 9 + \left(5 a + 2\right)\cdot 13 + \left(12 a + 11\right)\cdot 13^{2} + \left(3 a + 10\right)\cdot 13^{3} + \left(3 a + 1\right)\cdot 13^{4} + \left(2 a + 6\right)\cdot 13^{5} + \left(12 a + 6\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 4 }$ $=$ $$2 a + 7 + \left(7 a + 10\right)\cdot 13 + 4\cdot 13^{2} + \left(9 a + 2\right)\cdot 13^{3} + \left(9 a + 1\right)\cdot 13^{4} + \left(10 a + 5\right)\cdot 13^{5} + 3\cdot 13^{6} +O(13^{7})$$ $r_{ 5 }$ $=$ $$10 a + 7 + \left(9 a + 9\right)\cdot 13 + \left(10 a + 9\right)\cdot 13^{2} + \left(6 a + 3\right)\cdot 13^{3} + \left(8 a + 12\right)\cdot 13^{4} + \left(10 a + 11\right)\cdot 13^{5} + \left(a + 9\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 6 }$ $=$ $$3 a + 4 + \left(3 a + 9\right)\cdot 13 + \left(2 a + 10\right)\cdot 13^{2} + \left(6 a + 12\right)\cdot 13^{3} + 4 a\cdot 13^{4} + \left(2 a + 1\right)\cdot 13^{5} + \left(11 a + 1\right)\cdot 13^{6} +O(13^{7})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.