# Properties

 Label 3.41e3_257.6t11.2c1 Dimension 3 Group $S_4\times C_2$ Conductor $41^{3} \cdot 257$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4\times C_2$ Conductor: $17712697= 41^{3} \cdot 257$ Artin number field: Splitting field of $f=x^{6} - 2 x^{5} + 18 x^{4} + 56 x^{3} - 257 x^{2} - 794 x - 1671$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4\times C_2$ Parity: Even Determinant: 1.41_257.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $x^{2} + 60 x + 2$
Roots: \begin{aligned} r_{ 1 } &= -63476863443391997279 +O\left(61^{ 12 }\right) \\ r_{ 2 } &= -400410767552906194475 a - 67483561892818165132 +O\left(61^{ 12 }\right) \\ r_{ 3 } &= 1075762725181747628611 a - 1212099706827032726355 +O\left(61^{ 12 }\right) \\ r_{ 4 } &= 674177269852201295279 +O\left(61^{ 12 }\right) \\ r_{ 5 } &= -1075762725181747628611 a + 600861139707763515010 +O\left(61^{ 12 }\right) \\ r_{ 6 } &= 400410767552906194475 a + 68021722603278078479 +O\left(61^{ 12 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $3$ $1$ $2$ $(1,4)(2,6)(3,5)$ $-3$ $3$ $2$ $(3,5)$ $1$ $3$ $2$ $(2,6)(3,5)$ $-1$ $6$ $2$ $(1,2)(4,6)$ $1$ $6$ $2$ $(1,2)(3,5)(4,6)$ $-1$ $8$ $3$ $(1,3,2)(4,5,6)$ $0$ $6$ $4$ $(2,3,6,5)$ $1$ $6$ $4$ $(1,4)(2,3,6,5)$ $-1$ $8$ $6$ $(1,3,6,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.