# Properties

 Label 2.2548.6t5.d.a Dimension $2$ Group $S_3\times C_3$ Conductor $2548$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$2548$$$$\medspace = 2^{2} \cdot 7^{2} \cdot 13$$ Artin stem field: Galois closure of 6.0.25969216.3 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.364.6t1.g.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.676.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2x^{5} - 12x^{4} + 46x^{3} + 23x^{2} - 240x + 225$$ x^6 - 2*x^5 - 12*x^4 + 46*x^3 + 23*x^2 - 240*x + 225 .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{2} + 18x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$7 a + 8 + \left(12 a + 12\right)\cdot 19 + \left(15 a + 18\right)\cdot 19^{2} + \left(10 a + 18\right)\cdot 19^{3} + \left(12 a + 17\right)\cdot 19^{4} + 4 a\cdot 19^{5} + 11\cdot 19^{6} +O(19^{7})$$ 7*a + 8 + (12*a + 12)*19 + (15*a + 18)*19^2 + (10*a + 18)*19^3 + (12*a + 17)*19^4 + 4*a*19^5 + 11*19^6+O(19^7) $r_{ 2 }$ $=$ $$12 a + 15 + \left(6 a + 17\right)\cdot 19 + \left(3 a + 2\right)\cdot 19^{2} + \left(8 a + 14\right)\cdot 19^{3} + 6 a\cdot 19^{4} + \left(14 a + 12\right)\cdot 19^{5} + \left(18 a + 6\right)\cdot 19^{6} +O(19^{7})$$ 12*a + 15 + (6*a + 17)*19 + (3*a + 2)*19^2 + (8*a + 14)*19^3 + 6*a*19^4 + (14*a + 12)*19^5 + (18*a + 6)*19^6+O(19^7) $r_{ 3 }$ $=$ $$2 a + 4 + \left(5 a + 17\right)\cdot 19 + \left(a + 7\right)\cdot 19^{2} + 19^{3} + \left(16 a + 17\right)\cdot 19^{4} + \left(13 a + 17\right)\cdot 19^{5} + \left(2 a + 11\right)\cdot 19^{6} +O(19^{7})$$ 2*a + 4 + (5*a + 17)*19 + (a + 7)*19^2 + 19^3 + (16*a + 17)*19^4 + (13*a + 17)*19^5 + (2*a + 11)*19^6+O(19^7) $r_{ 4 }$ $=$ $$5 a + 1 + \left(7 a + 3\right)\cdot 19 + 15\cdot 19^{2} + \left(18 a + 11\right)\cdot 19^{3} + \left(16 a + 13\right)\cdot 19^{4} + \left(11 a + 7\right)\cdot 19^{5} + \left(a + 18\right)\cdot 19^{6} +O(19^{7})$$ 5*a + 1 + (7*a + 3)*19 + 15*19^2 + (18*a + 11)*19^3 + (16*a + 13)*19^4 + (11*a + 7)*19^5 + (a + 18)*19^6+O(19^7) $r_{ 5 }$ $=$ $$17 a + 6 + \left(13 a + 1\right)\cdot 19 + \left(17 a + 4\right)\cdot 19^{2} + 18 a\cdot 19^{3} + \left(2 a + 14\right)\cdot 19^{4} + \left(5 a + 15\right)\cdot 19^{5} + 16 a\cdot 19^{6} +O(19^{7})$$ 17*a + 6 + (13*a + 1)*19 + (17*a + 4)*19^2 + 18*a*19^3 + (2*a + 14)*19^4 + (5*a + 15)*19^5 + 16*a*19^6+O(19^7) $r_{ 6 }$ $=$ $$14 a + 6 + \left(11 a + 5\right)\cdot 19 + \left(18 a + 8\right)\cdot 19^{2} + 10\cdot 19^{3} + \left(2 a + 12\right)\cdot 19^{4} + \left(7 a + 2\right)\cdot 19^{5} + \left(17 a + 8\right)\cdot 19^{6} +O(19^{7})$$ 14*a + 6 + (11*a + 5)*19 + (18*a + 8)*19^2 + 10*19^3 + (2*a + 12)*19^4 + (7*a + 2)*19^5 + (17*a + 8)*19^6+O(19^7)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.