# Properties

 Label 3.41e3_257e2.6t11.2c1 Dimension 3 Group $S_4\times C_2$ Conductor $41^{3} \cdot 257^{2}$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $3$ Group: $S_4\times C_2$ Conductor: $4552163129= 41^{3} \cdot 257^{2}$ 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: 1.41.2t1.1c1

## 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:
 $r_{ 1 }$ $=$ $4 + 37\cdot 61 + 43\cdot 61^{2} + 14\cdot 61^{3} + 56\cdot 61^{4} + 52\cdot 61^{5} + 34\cdot 61^{6} + 35\cdot 61^{7} + 55\cdot 61^{8} + 33\cdot 61^{10} + 59\cdot 61^{11} +O\left(61^{ 12 }\right)$ $r_{ 2 }$ $=$ $49 a + 51 + \left(34 a + 50\right)\cdot 61 + \left(52 a + 59\right)\cdot 61^{2} + \left(45 a + 60\right)\cdot 61^{3} + \left(54 a + 20\right)\cdot 61^{4} + \left(6 a + 56\right)\cdot 61^{5} + \left(5 a + 36\right)\cdot 61^{6} + \left(17 a + 30\right)\cdot 61^{7} + \left(44 a + 17\right)\cdot 61^{8} + \left(41 a + 24\right)\cdot 61^{9} + \left(48 a + 27\right)\cdot 61^{10} + \left(51 a + 59\right)\cdot 61^{11} +O\left(61^{ 12 }\right)$ $r_{ 3 }$ $=$ $59 a + 9 + \left(57 a + 17\right)\cdot 61 + \left(43 a + 3\right)\cdot 61^{2} + \left(36 a + 41\right)\cdot 61^{3} + \left(5 a + 39\right)\cdot 61^{4} + \left(57 a + 8\right)\cdot 61^{5} + \left(25 a + 4\right)\cdot 61^{6} + \left(4 a + 22\right)\cdot 61^{7} + \left(34 a + 53\right)\cdot 61^{8} + \left(3 a + 49\right)\cdot 61^{9} + \left(44 a + 8\right)\cdot 61^{10} + \left(24 a + 33\right)\cdot 61^{11} +O\left(61^{ 12 }\right)$ $r_{ 4 }$ $=$ $14 + 25\cdot 61 + 9\cdot 61^{2} + 39\cdot 61^{3} + 27\cdot 61^{4} + 57\cdot 61^{5} + 37\cdot 61^{6} + 51\cdot 61^{7} + 50\cdot 61^{8} + 5\cdot 61^{9} + 30\cdot 61^{10} + 15\cdot 61^{11} +O\left(61^{ 12 }\right)$ $r_{ 5 }$ $=$ $2 a + 7 + \left(3 a + 16\right)\cdot 61 + \left(17 a + 50\right)\cdot 61^{2} + \left(24 a + 33\right)\cdot 61^{3} + \left(55 a + 8\right)\cdot 61^{4} + \left(3 a + 60\right)\cdot 61^{5} + \left(35 a + 33\right)\cdot 61^{6} + 56 a\cdot 61^{7} + \left(26 a + 22\right)\cdot 61^{8} + \left(57 a + 19\right)\cdot 61^{9} + \left(16 a + 49\right)\cdot 61^{10} + \left(36 a + 13\right)\cdot 61^{11} +O\left(61^{ 12 }\right)$ $r_{ 6 }$ $=$ $12 a + 39 + \left(26 a + 36\right)\cdot 61 + \left(8 a + 16\right)\cdot 61^{2} + \left(15 a + 54\right)\cdot 61^{3} + \left(6 a + 29\right)\cdot 61^{4} + \left(54 a + 8\right)\cdot 61^{5} + \left(55 a + 35\right)\cdot 61^{6} + \left(43 a + 42\right)\cdot 61^{7} + \left(16 a + 44\right)\cdot 61^{8} + \left(19 a + 21\right)\cdot 61^{9} + \left(12 a + 34\right)\cdot 61^{10} + \left(9 a + 1\right)\cdot 61^{11} +O\left(61^{ 12 }\right)$

### 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.