# 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: \begin{aligned} r_{ 1 } &= -488519553380745 a + 388251348590356 +O\left(47^{ 9 }\right) \\ r_{ 2 } &= -290687329545959 a + 133242669077364 +O\left(47^{ 9 }\right) \\ r_{ 3 } &= -377273137258896 +O\left(47^{ 9 }\right) \\ r_{ 4 } &= 488519553380745 a - 10978211331459 +O\left(47^{ 9 }\right) \\ r_{ 5 } &= 290687329545959 a - 215393178587685 +O\left(47^{ 9 }\right) \\ r_{ 6 } &= 82150509510322 +O\left(47^{ 9 }\right) \\ \end{aligned}

### 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.