# Properties

 Label 2.280.12t18.b Dimension $2$ Group $C_6\times S_3$ Conductor $280$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_6\times S_3$ Conductor: $$280$$$$\medspace = 2^{3} \cdot 5 \cdot 7$$ Artin number field: Galois closure of 12.0.9834496000000.1 Galois orbit size: $2$ Smallest permutation container: $C_6\times S_3$ Parity: odd Projective image: $S_3$ Projective field: 3.1.1960.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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