# Properties

 Label 2.1368.12t18.b.b Dimension $2$ Group $C_6\times S_3$ Conductor $1368$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_6\times S_3$ Conductor: $$1368$$$$\medspace = 2^{3} \cdot 3^{2} \cdot 19$$ Artin stem field: Galois closure of 12.0.24904730935296.2 Galois orbit size: $2$ Smallest permutation container: $C_6\times S_3$ Parity: odd Determinant: 1.152.6t1.c.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.2888.1

## Defining polynomial

 $f(x)$ $=$ $$x^{12} - 2x^{11} + 3x^{10} - 6x^{9} + 5x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 5x^{4} - 12x^{3} + 16x^{2} - 12x + 9$$ x^12 - 2*x^11 + 3*x^10 - 6*x^9 + 5*x^8 - 2*x^7 + 2*x^6 + 2*x^5 + 5*x^4 - 12*x^3 + 16*x^2 - 12*x + 9 .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{6} + 10x^{3} + 11x^{2} + 11x + 2$$

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

## Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

 Cycle notation $(1,4,2,5,3,6)(7,11,8,12,9,10)$ $(1,10)(2,11)(3,12)(4,7)(5,8)(6,9)$ $(4,6,5)(7,9,8)$ $(1,11,3,10,2,12)(4,8,6,7,5,9)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 12 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,10)(2,11)(3,12)(4,7)(5,8)(6,9)$ $-2$ $3$ $2$ $(1,5)(2,6)(3,4)(7,12)(8,10)(9,11)$ $0$ $3$ $2$ $(1,8)(2,9)(3,7)(4,12)(5,10)(6,11)$ $0$ $1$ $3$ $(1,2,3)(4,5,6)(7,8,9)(10,11,12)$ $-2 \zeta_{3} - 2$ $1$ $3$ $(1,3,2)(4,6,5)(7,9,8)(10,12,11)$ $2 \zeta_{3}$ $2$ $3$ $(4,6,5)(7,9,8)$ $\zeta_{3} + 1$ $2$ $3$ $(4,5,6)(7,8,9)$ $-\zeta_{3}$ $2$ $3$ $(1,2,3)(4,6,5)(7,9,8)(10,11,12)$ $-1$ $1$ $6$ $(1,11,3,10,2,12)(4,8,6,7,5,9)$ $2 \zeta_{3} + 2$ $1$ $6$ $(1,12,2,10,3,11)(4,9,5,7,6,8)$ $-2 \zeta_{3}$ $2$ $6$ $(1,10)(2,11)(3,12)(4,9,5,7,6,8)$ $-\zeta_{3} - 1$ $2$ $6$ $(1,10)(2,11)(3,12)(4,8,6,7,5,9)$ $\zeta_{3}$ $2$ $6$ $(1,12,2,10,3,11)(4,8,6,7,5,9)$ $1$ $3$ $6$ $(1,4,2,5,3,6)(7,11,8,12,9,10)$ $0$ $3$ $6$ $(1,6,3,5,2,4)(7,10,9,12,8,11)$ $0$ $3$ $6$ $(1,7,2,8,3,9)(4,11,5,12,6,10)$ $0$ $3$ $6$ $(1,9,3,8,2,7)(4,10,6,12,5,11)$ $0$

The blue line marks the conjugacy class containing complex conjugation.