# Properties

 Label 2.41e2_257.6t3.2c1 Dimension 2 Group $D_{6}$ Conductor $41^{2} \cdot 257$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $432017= 41^{2} \cdot 257$ Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - 267 x^{4} + 504 x^{3} + 17720 x^{2} - 31624 x - 249501$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{6}$ Parity: Even Determinant: 1.257.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$
Roots:
 $r_{ 1 }$ $=$ $40 a + 8 + \left(17 a + 46\right)\cdot 47 + \left(43 a + 17\right)\cdot 47^{2} + \left(40 a + 30\right)\cdot 47^{3} + \left(44 a + 13\right)\cdot 47^{4} + \left(22 a + 24\right)\cdot 47^{5} + \left(34 a + 16\right)\cdot 47^{6} + \left(22 a + 14\right)\cdot 47^{7} + \left(26 a + 16\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ $r_{ 2 }$ $=$ $38 a + 17 + \left(a + 27\right)\cdot 47 + \left(38 a + 30\right)\cdot 47^{2} + \left(8 a + 26\right)\cdot 47^{3} + \left(26 a + 20\right)\cdot 47^{4} + \left(28 a + 3\right)\cdot 47^{5} + 10 a\cdot 47^{6} + \left(37 a + 28\right)\cdot 47^{7} + \left(34 a + 5\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ $r_{ 3 }$ $=$ $46 + 5\cdot 47 + 36\cdot 47^{2} + 41\cdot 47^{3} + 17\cdot 47^{4} + 44\cdot 47^{5} + 14\cdot 47^{6} + 7\cdot 47^{7} + 31\cdot 47^{8} +O\left(47^{ 9 }\right)$ $r_{ 4 }$ $=$ $7 a + 41 + \left(29 a + 41\right)\cdot 47 + \left(3 a + 39\right)\cdot 47^{2} + \left(6 a + 21\right)\cdot 47^{3} + \left(2 a + 15\right)\cdot 47^{4} + \left(24 a + 25\right)\cdot 47^{5} + \left(12 a + 15\right)\cdot 47^{6} + \left(24 a + 25\right)\cdot 47^{7} + \left(20 a + 46\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ $r_{ 5 }$ $=$ $9 a + 46 + \left(45 a + 39\right)\cdot 47 + \left(8 a + 10\right)\cdot 47^{2} + \left(38 a + 6\right)\cdot 47^{3} + \left(20 a + 17\right)\cdot 47^{4} + \left(18 a + 34\right)\cdot 47^{5} + \left(36 a + 39\right)\cdot 47^{6} + \left(9 a + 44\right)\cdot 47^{7} + \left(12 a + 37\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ $r_{ 6 }$ $=$ $32 + 26\cdot 47 + 5\cdot 47^{2} + 14\cdot 47^{3} + 9\cdot 47^{4} + 9\cdot 47^{5} + 7\cdot 47^{6} + 21\cdot 47^{7} + 3\cdot 47^{8} +O\left(47^{ 9 }\right)$

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

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

### Character values on conjugacy classes

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