# Properties

 Label 3.18571.6t6.a.a Dimension 3 Group $A_4\times C_2$ Conductor $7^{2} \cdot 379$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $3$ Group: $A_4\times C_2$ Conductor: $18571= 7^{2} \cdot 379$ Artin number field: Splitting field of 6.0.909979.1 defined by $f= x^{6} - 2 x^{5} + 7 x^{4} - 12 x^{3} + 21 x^{2} - 15 x + 13$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $A_4\times C_2$ Parity: Odd Determinant: 1.379.2t1.a.a Projective image: $A_4$ Projective field: Galois closure of 4.4.7038409.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $x^{2} + 38 x + 6$
Roots:
 $r_{ 1 }$ $=$ $17 a + 23 + \left(17 a + 38\right)\cdot 41 + \left(36 a + 13\right)\cdot 41^{2} + \left(21 a + 19\right)\cdot 41^{3} + \left(11 a + 8\right)\cdot 41^{4} + \left(26 a + 8\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ $r_{ 2 }$ $=$ $24 a + 33 + \left(23 a + 32\right)\cdot 41 + \left(4 a + 23\right)\cdot 41^{2} + \left(19 a + 7\right)\cdot 41^{3} + \left(29 a + 21\right)\cdot 41^{4} + \left(14 a + 34\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ $r_{ 3 }$ $=$ $39 a + 39 + \left(3 a + 25\right)\cdot 41 + \left(23 a + 2\right)\cdot 41^{2} + \left(37 a + 18\right)\cdot 41^{3} + \left(33 a + 9\right)\cdot 41^{4} + \left(17 a + 17\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ $r_{ 4 }$ $=$ $23 + 31\cdot 41 + 7\cdot 41^{2} + 36\cdot 41^{3} + 25\cdot 41^{4} + 18\cdot 41^{5} +O\left(41^{ 6 }\right)$ $r_{ 5 }$ $=$ $15 + 36\cdot 41 + 6\cdot 41^{2} + 16\cdot 41^{3} + 25\cdot 41^{4} + 7\cdot 41^{5} +O\left(41^{ 6 }\right)$ $r_{ 6 }$ $=$ $2 a + 33 + \left(37 a + 39\right)\cdot 41 + \left(17 a + 26\right)\cdot 41^{2} + \left(3 a + 25\right)\cdot 41^{3} + \left(7 a + 32\right)\cdot 41^{4} + \left(23 a + 36\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$

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

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

### Character values on conjugacy classes

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