# Properties

 Label 2.57.6t5.a.a Dimension $2$ Group $S_3\times C_3$ Conductor $57$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$57$$$$\medspace = 3 \cdot 19$$ Artin stem field: 6.0.9747.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.57.6t1.a.b Projective image: $S_3$ Projective stem field: 3.1.1083.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7 x + 2$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.