# Properties

 Label 2.3328.12t11.a Dimension $2$ Group $S_3 \times C_4$ Conductor $3328$ Indicator $0$

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## Basic invariants

 Dimension: $2$ Group: $S_3 \times C_4$ Conductor: $$3328$$$$\medspace = 2^{8} \cdot 13$$ Artin number field: Galois closure of 12.0.981348487528448.5 Galois orbit size: $2$ Smallest permutation container: $S_3 \times C_4$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.104.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{4} + 2x^{2} + 11x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$4 a^{3} + 14 a^{2} + 8 a + 9 + \left(3 a^{3} + 10 a^{2} + 13 a + 13\right)\cdot 19 + \left(12 a^{3} + 2 a^{2} + 16 a + 12\right)\cdot 19^{2} + \left(8 a^{3} + 10 a^{2} + 6 a\right)\cdot 19^{3} + \left(18 a^{3} + 16 a^{2} + 2 a + 12\right)\cdot 19^{4} + \left(8 a^{3} + 14 a^{2} + 17 a + 17\right)\cdot 19^{5} + \left(3 a^{3} + 6 a^{2} + 3 a + 6\right)\cdot 19^{6} + \left(12 a^{3} + 15 a^{2} + a + 11\right)\cdot 19^{7} + \left(16 a^{3} + 12 a^{2} + 7\right)\cdot 19^{8} +O(19^{9})$$ 4*a^3 + 14*a^2 + 8*a + 9 + (3*a^3 + 10*a^2 + 13*a + 13)*19 + (12*a^3 + 2*a^2 + 16*a + 12)*19^2 + (8*a^3 + 10*a^2 + 6*a)*19^3 + (18*a^3 + 16*a^2 + 2*a + 12)*19^4 + (8*a^3 + 14*a^2 + 17*a + 17)*19^5 + (3*a^3 + 6*a^2 + 3*a + 6)*19^6 + (12*a^3 + 15*a^2 + a + 11)*19^7 + (16*a^3 + 12*a^2 + 7)*19^8+O(19^9) $r_{ 2 }$ $=$ $$17 a^{3} + 18 a^{2} + 6 a + 11 + \left(7 a^{3} + 15 a + 18\right)\cdot 19 + \left(13 a^{2} + 9 a + 6\right)\cdot 19^{2} + \left(12 a^{3} + 4 a^{2} + 5 a + 18\right)\cdot 19^{3} + \left(5 a^{3} + 10 a^{2} + 7 a + 13\right)\cdot 19^{4} + \left(2 a + 7\right)\cdot 19^{5} + \left(11 a^{3} + 13 a^{2} + 10 a + 18\right)\cdot 19^{6} + \left(9 a^{3} + 11 a^{2} + 15 a + 9\right)\cdot 19^{7} + \left(17 a^{3} + 11 a^{2} + a + 13\right)\cdot 19^{8} +O(19^{9})$$ 17*a^3 + 18*a^2 + 6*a + 11 + (7*a^3 + 15*a + 18)*19 + (13*a^2 + 9*a + 6)*19^2 + (12*a^3 + 4*a^2 + 5*a + 18)*19^3 + (5*a^3 + 10*a^2 + 7*a + 13)*19^4 + (2*a + 7)*19^5 + (11*a^3 + 13*a^2 + 10*a + 18)*19^6 + (9*a^3 + 11*a^2 + 15*a + 9)*19^7 + (17*a^3 + 11*a^2 + a + 13)*19^8+O(19^9) $r_{ 3 }$ $=$ $$9 a^{3} + 17 a^{2} + 16 a + 1 + \left(5 a^{3} + 12 a^{2} + 18 a + 1\right)\cdot 19 + \left(15 a^{3} + 6 a^{2} + 10 a + 14\right)\cdot 19^{2} + \left(13 a^{3} + 13 a^{2} + 8 a + 3\right)\cdot 19^{3} + \left(6 a^{3} + 17 a^{2} + 8 a + 16\right)\cdot 19^{4} + \left(2 a^{3} + 6 a^{2} + 15 a + 16\right)\cdot 19^{5} + \left(13 a^{3} + 14 a^{2} + 18 a + 3\right)\cdot 19^{6} + \left(2 a^{3} + 17 a^{2} + 5 a + 16\right)\cdot 19^{7} + \left(17 a^{3} + 8 a^{2} + 12 a + 7\right)\cdot 19^{8} +O(19^{9})$$ 9*a^3 + 17*a^2 + 16*a + 1 + (5*a^3 + 12*a^2 + 18*a + 1)*19 + (15*a^3 + 6*a^2 + 10*a + 14)*19^2 + (13*a^3 + 13*a^2 + 8*a + 3)*19^3 + (6*a^3 + 17*a^2 + 8*a + 16)*19^4 + (2*a^3 + 6*a^2 + 15*a + 16)*19^5 + (13*a^3 + 14*a^2 + 18*a + 3)*19^6 + (2*a^3 + 17*a^2 + 5*a + 16)*19^7 + (17*a^3 + 8*a^2 + 12*a + 7)*19^8+O(19^9) $r_{ 4 }$ $=$ $$17 a^{3} + a^{2} + 3 a + 13 + \left(12 a^{3} + 9 a^{2} + 7 a + 15\right)\cdot 19 + \left(18 a^{3} + 6 a^{2} + 6 a + 3\right)\cdot 19^{2} + \left(14 a^{3} + 16 a^{2} + 9 a + 2\right)\cdot 19^{3} + \left(5 a^{3} + 6 a^{2} + 3 a + 7\right)\cdot 19^{4} + \left(15 a^{3} + 7 a^{2} + 12 a + 5\right)\cdot 19^{5} + \left(16 a^{3} + 6 a^{2} + 14 a + 7\right)\cdot 19^{6} + \left(13 a^{2} + a + 6\right)\cdot 19^{7} + \left(9 a^{3} + 13 a + 18\right)\cdot 19^{8} +O(19^{9})$$ 17*a^3 + a^2 + 3*a + 13 + (12*a^3 + 9*a^2 + 7*a + 15)*19 + (18*a^3 + 6*a^2 + 6*a + 3)*19^2 + (14*a^3 + 16*a^2 + 9*a + 2)*19^3 + (5*a^3 + 6*a^2 + 3*a + 7)*19^4 + (15*a^3 + 7*a^2 + 12*a + 5)*19^5 + (16*a^3 + 6*a^2 + 14*a + 7)*19^6 + (13*a^2 + a + 6)*19^7 + (9*a^3 + 13*a + 18)*19^8+O(19^9) $r_{ 5 }$ $=$ $$6 a^{3} + a^{2} + 4 a + 3 + \left(9 a^{3} + 12 a^{2} + 13 a + 8\right)\cdot 19 + \left(15 a^{3} + 10 a^{2} + 17 a + 5\right)\cdot 19^{2} + \left(16 a^{3} + 13 a^{2} + 10 a\right)\cdot 19^{3} + \left(13 a^{3} + 3 a + 6\right)\cdot 19^{4} + \left(17 a^{3} + 8 a^{2} + 7 a + 2\right)\cdot 19^{5} + \left(12 a^{3} + 4 a^{2} + 5 a + 16\right)\cdot 19^{6} + \left(10 a^{3} + 16 a^{2} + 18 a + 18\right)\cdot 19^{7} + \left(6 a^{3} + 6 a^{2} + 8 a + 3\right)\cdot 19^{8} +O(19^{9})$$ 6*a^3 + a^2 + 4*a + 3 + (9*a^3 + 12*a^2 + 13*a + 8)*19 + (15*a^3 + 10*a^2 + 17*a + 5)*19^2 + (16*a^3 + 13*a^2 + 10*a)*19^3 + (13*a^3 + 3*a + 6)*19^4 + (17*a^3 + 8*a^2 + 7*a + 2)*19^5 + (12*a^3 + 4*a^2 + 5*a + 16)*19^6 + (10*a^3 + 16*a^2 + 18*a + 18)*19^7 + (6*a^3 + 6*a^2 + 8*a + 3)*19^8+O(19^9) $r_{ 6 }$ $=$ $$2 a^{3} + 16 a^{2} + 4 + \left(9 a^{3} + a^{2} + 2 a + 15\right)\cdot 19 + \left(13 a^{3} + 15 a^{2} + 6 a + 2\right)\cdot 19^{2} + \left(8 a^{3} + 12 a^{2} + 15 a + 18\right)\cdot 19^{3} + \left(10 a^{3} + 16 a^{2} + 9 a + 7\right)\cdot 19^{4} + \left(a^{2} + 16 a + 6\right)\cdot 19^{5} + \left(3 a^{3} + 17 a^{2} + a + 18\right)\cdot 19^{6} + \left(18 a^{3} + 4 a^{2} + 9 a + 16\right)\cdot 19^{7} + \left(2 a^{3} + 9 a^{2} + 3 a + 9\right)\cdot 19^{8} +O(19^{9})$$ 2*a^3 + 16*a^2 + 4 + (9*a^3 + a^2 + 2*a + 15)*19 + (13*a^3 + 15*a^2 + 6*a + 2)*19^2 + (8*a^3 + 12*a^2 + 15*a + 18)*19^3 + (10*a^3 + 16*a^2 + 9*a + 7)*19^4 + (a^2 + 16*a + 6)*19^5 + (3*a^3 + 17*a^2 + a + 18)*19^6 + (18*a^3 + 4*a^2 + 9*a + 16)*19^7 + (2*a^3 + 9*a^2 + 3*a + 9)*19^8+O(19^9) $r_{ 7 }$ $=$ $$15 a^{3} + 5 a^{2} + 11 a + 10 + \left(15 a^{3} + 8 a^{2} + 5 a + 5\right)\cdot 19 + \left(6 a^{3} + 16 a^{2} + 2 a + 6\right)\cdot 19^{2} + \left(10 a^{3} + 8 a^{2} + 12 a + 18\right)\cdot 19^{3} + \left(2 a^{2} + 16 a + 6\right)\cdot 19^{4} + \left(10 a^{3} + 4 a^{2} + a + 1\right)\cdot 19^{5} + \left(15 a^{3} + 12 a^{2} + 15 a + 12\right)\cdot 19^{6} + \left(6 a^{3} + 3 a^{2} + 17 a + 7\right)\cdot 19^{7} + \left(2 a^{3} + 6 a^{2} + 18 a + 11\right)\cdot 19^{8} +O(19^{9})$$ 15*a^3 + 5*a^2 + 11*a + 10 + (15*a^3 + 8*a^2 + 5*a + 5)*19 + (6*a^3 + 16*a^2 + 2*a + 6)*19^2 + (10*a^3 + 8*a^2 + 12*a + 18)*19^3 + (2*a^2 + 16*a + 6)*19^4 + (10*a^3 + 4*a^2 + a + 1)*19^5 + (15*a^3 + 12*a^2 + 15*a + 12)*19^6 + (6*a^3 + 3*a^2 + 17*a + 7)*19^7 + (2*a^3 + 6*a^2 + 18*a + 11)*19^8+O(19^9) $r_{ 8 }$ $=$ $$2 a^{3} + a^{2} + 13 a + 8 + \left(11 a^{3} + 18 a^{2} + 3 a\right)\cdot 19 + \left(18 a^{3} + 5 a^{2} + 9 a + 12\right)\cdot 19^{2} + \left(6 a^{3} + 14 a^{2} + 13 a\right)\cdot 19^{3} + \left(13 a^{3} + 8 a^{2} + 11 a + 5\right)\cdot 19^{4} + \left(18 a^{3} + 18 a^{2} + 16 a + 11\right)\cdot 19^{5} + \left(7 a^{3} + 5 a^{2} + 8 a\right)\cdot 19^{6} + \left(9 a^{3} + 7 a^{2} + 3 a + 9\right)\cdot 19^{7} + \left(a^{3} + 7 a^{2} + 17 a + 5\right)\cdot 19^{8} +O(19^{9})$$ 2*a^3 + a^2 + 13*a + 8 + (11*a^3 + 18*a^2 + 3*a)*19 + (18*a^3 + 5*a^2 + 9*a + 12)*19^2 + (6*a^3 + 14*a^2 + 13*a)*19^3 + (13*a^3 + 8*a^2 + 11*a + 5)*19^4 + (18*a^3 + 18*a^2 + 16*a + 11)*19^5 + (7*a^3 + 5*a^2 + 8*a)*19^6 + (9*a^3 + 7*a^2 + 3*a + 9)*19^7 + (a^3 + 7*a^2 + 17*a + 5)*19^8+O(19^9) $r_{ 9 }$ $=$ $$10 a^{3} + 2 a^{2} + 3 a + 18 + \left(13 a^{3} + 6 a^{2} + 17\right)\cdot 19 + \left(3 a^{3} + 12 a^{2} + 8 a + 4\right)\cdot 19^{2} + \left(5 a^{3} + 5 a^{2} + 10 a + 15\right)\cdot 19^{3} + \left(12 a^{3} + a^{2} + 10 a + 2\right)\cdot 19^{4} + \left(16 a^{3} + 12 a^{2} + 3 a + 2\right)\cdot 19^{5} + \left(5 a^{3} + 4 a^{2} + 15\right)\cdot 19^{6} + \left(16 a^{3} + a^{2} + 13 a + 2\right)\cdot 19^{7} + \left(a^{3} + 10 a^{2} + 6 a + 11\right)\cdot 19^{8} +O(19^{9})$$ 10*a^3 + 2*a^2 + 3*a + 18 + (13*a^3 + 6*a^2 + 17)*19 + (3*a^3 + 12*a^2 + 8*a + 4)*19^2 + (5*a^3 + 5*a^2 + 10*a + 15)*19^3 + (12*a^3 + a^2 + 10*a + 2)*19^4 + (16*a^3 + 12*a^2 + 3*a + 2)*19^5 + (5*a^3 + 4*a^2 + 15)*19^6 + (16*a^3 + a^2 + 13*a + 2)*19^7 + (a^3 + 10*a^2 + 6*a + 11)*19^8+O(19^9) $r_{ 10 }$ $=$ $$2 a^{3} + 18 a^{2} + 16 a + 6 + \left(6 a^{3} + 9 a^{2} + 11 a + 3\right)\cdot 19 + \left(12 a^{2} + 12 a + 15\right)\cdot 19^{2} + \left(4 a^{3} + 2 a^{2} + 9 a + 16\right)\cdot 19^{3} + \left(13 a^{3} + 12 a^{2} + 15 a + 11\right)\cdot 19^{4} + \left(3 a^{3} + 11 a^{2} + 6 a + 13\right)\cdot 19^{5} + \left(2 a^{3} + 12 a^{2} + 4 a + 11\right)\cdot 19^{6} + \left(18 a^{3} + 5 a^{2} + 17 a + 12\right)\cdot 19^{7} + \left(9 a^{3} + 18 a^{2} + 5 a\right)\cdot 19^{8} +O(19^{9})$$ 2*a^3 + 18*a^2 + 16*a + 6 + (6*a^3 + 9*a^2 + 11*a + 3)*19 + (12*a^2 + 12*a + 15)*19^2 + (4*a^3 + 2*a^2 + 9*a + 16)*19^3 + (13*a^3 + 12*a^2 + 15*a + 11)*19^4 + (3*a^3 + 11*a^2 + 6*a + 13)*19^5 + (2*a^3 + 12*a^2 + 4*a + 11)*19^6 + (18*a^3 + 5*a^2 + 17*a + 12)*19^7 + (9*a^3 + 18*a^2 + 5*a)*19^8+O(19^9) $r_{ 11 }$ $=$ $$13 a^{3} + 18 a^{2} + 15 a + 16 + \left(9 a^{3} + 6 a^{2} + 5 a + 10\right)\cdot 19 + \left(3 a^{3} + 8 a^{2} + a + 13\right)\cdot 19^{2} + \left(2 a^{3} + 5 a^{2} + 8 a + 18\right)\cdot 19^{3} + \left(5 a^{3} + 18 a^{2} + 15 a + 12\right)\cdot 19^{4} + \left(a^{3} + 10 a^{2} + 11 a + 16\right)\cdot 19^{5} + \left(6 a^{3} + 14 a^{2} + 13 a + 2\right)\cdot 19^{6} + \left(8 a^{3} + 2 a^{2}\right)\cdot 19^{7} + \left(12 a^{3} + 12 a^{2} + 10 a + 15\right)\cdot 19^{8} +O(19^{9})$$ 13*a^3 + 18*a^2 + 15*a + 16 + (9*a^3 + 6*a^2 + 5*a + 10)*19 + (3*a^3 + 8*a^2 + a + 13)*19^2 + (2*a^3 + 5*a^2 + 8*a + 18)*19^3 + (5*a^3 + 18*a^2 + 15*a + 12)*19^4 + (a^3 + 10*a^2 + 11*a + 16)*19^5 + (6*a^3 + 14*a^2 + 13*a + 2)*19^6 + (8*a^3 + 2*a^2)*19^7 + (12*a^3 + 12*a^2 + 10*a + 15)*19^8+O(19^9) $r_{ 12 }$ $=$ $$17 a^{3} + 3 a^{2} + 15 + \left(9 a^{3} + 17 a^{2} + 17 a + 3\right)\cdot 19 + \left(5 a^{3} + 3 a^{2} + 12 a + 16\right)\cdot 19^{2} + \left(10 a^{3} + 6 a^{2} + 3 a\right)\cdot 19^{3} + \left(8 a^{3} + 2 a^{2} + 9 a + 11\right)\cdot 19^{4} + \left(18 a^{3} + 17 a^{2} + 2 a + 12\right)\cdot 19^{5} + \left(15 a^{3} + a^{2} + 17 a\right)\cdot 19^{6} + \left(14 a^{2} + 9 a + 2\right)\cdot 19^{7} + \left(16 a^{3} + 9 a^{2} + 15 a + 9\right)\cdot 19^{8} +O(19^{9})$$ 17*a^3 + 3*a^2 + 15 + (9*a^3 + 17*a^2 + 17*a + 3)*19 + (5*a^3 + 3*a^2 + 12*a + 16)*19^2 + (10*a^3 + 6*a^2 + 3*a)*19^3 + (8*a^3 + 2*a^2 + 9*a + 11)*19^4 + (18*a^3 + 17*a^2 + 2*a + 12)*19^5 + (15*a^3 + a^2 + 17*a)*19^6 + (14*a^2 + 9*a + 2)*19^7 + (16*a^3 + 9*a^2 + 15*a + 9)*19^8+O(19^9)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 12 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ $-2$ $-2$ $3$ $2$ $(2,5)(3,12)(6,9)(8,11)$ $0$ $0$ $3$ $2$ $(1,7)(2,11)(3,6)(4,10)(5,8)(9,12)$ $0$ $0$ $2$ $3$ $(1,11,8)(2,7,5)(3,10,12)(4,6,9)$ $-1$ $-1$ $1$ $4$ $(1,10,7,4)(2,9,8,3)(5,6,11,12)$ $-2 \zeta_{4}$ $2 \zeta_{4}$ $1$ $4$ $(1,4,7,10)(2,3,8,9)(5,12,11,6)$ $2 \zeta_{4}$ $-2 \zeta_{4}$ $3$ $4$ $(1,10,7,4)(2,6,8,12)(3,5,9,11)$ $0$ $0$ $3$ $4$ $(1,4,7,10)(2,12,8,6)(3,11,9,5)$ $0$ $0$ $2$ $6$ $(1,2,11,7,8,5)(3,6,10,9,12,4)$ $1$ $1$ $2$ $12$ $(1,6,2,10,11,9,7,12,8,4,5,3)$ $-\zeta_{4}$ $\zeta_{4}$ $2$ $12$ $(1,12,2,4,11,3,7,6,8,10,5,9)$ $\zeta_{4}$ $-\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.