# Properties

 Label 2.140.6t5.a Dimension $2$ Group $S_3\times C_3$ Conductor $140$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$140$$$$\medspace = 2^{2} \cdot 5 \cdot 7$$ Artin number field: Galois closure of 6.0.392000.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ 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 $\Q_{ 13 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{2} + 12 x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$12 a + 6 + \left(7 a + 4\right)\cdot 13 + \left(9 a + 1\right)\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(7 a + 3\right)\cdot 13^{4} + \left(11 a + 10\right)\cdot 13^{5} + \left(6 a + 2\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 2 }$ $=$ $$7 a + 11 + \left(10 a + 8\right)\cdot 13 + \left(3 a + 8\right)\cdot 13^{2} + 9\cdot 13^{3} + \left(3 a + 7\right)\cdot 13^{4} + \left(2 a + 7\right)\cdot 13^{5} + \left(12 a + 12\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 3 }$ $=$ $$a + 5 + 5 a\cdot 13 + \left(3 a + 3\right)\cdot 13^{2} + \left(3 a + 8\right)\cdot 13^{3} + \left(5 a + 1\right)\cdot 13^{4} + \left(a + 1\right)\cdot 13^{5} + \left(6 a + 11\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 4 }$ $=$ $$6 a + 5 + \left(2 a + 12\right)\cdot 13 + \left(9 a + 1\right)\cdot 13^{2} + \left(12 a + 6\right)\cdot 13^{3} + \left(9 a + 10\right)\cdot 13^{4} + \left(10 a + 6\right)\cdot 13^{5} + 9\cdot 13^{6} +O(13^{7})$$ $r_{ 5 }$ $=$ $$8 a + 3 + \left(2 a + 9\right)\cdot 13 + \left(7 a + 9\right)\cdot 13^{2} + \left(3 a + 11\right)\cdot 13^{3} + \left(8 a + 11\right)\cdot 13^{4} + \left(3 a + 8\right)\cdot 13^{5} + 5 a\cdot 13^{6} +O(13^{7})$$ $r_{ 6 }$ $=$ $$5 a + 11 + \left(10 a + 3\right)\cdot 13 + \left(5 a + 1\right)\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(4 a + 3\right)\cdot 13^{4} + \left(9 a + 4\right)\cdot 13^{5} + \left(7 a + 2\right)\cdot 13^{6} +O(13^{7})$$

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

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

### Character values on conjugacy classes

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