# Properties

 Label 2.140.12t18.a Dimension $2$ Group $C_6\times S_3$ Conductor $140$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_6\times S_3$ Conductor: $$140$$$$\medspace = 2^{2} \cdot 5 \cdot 7$$ Artin number field: Galois closure of 12.0.153664000000.1 Galois orbit size: $2$ Smallest permutation container: $C_6\times S_3$ Parity: odd Projective image: $S_3$ Projective field: 3.1.980.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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