# Properties

 Label 3.29e3_229.6t11.2c1 Dimension 3 Group $S_4\times C_2$ Conductor $29^{3} \cdot 229$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4\times C_2$ Conductor: $5585081= 29^{3} \cdot 229$ Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - x^{4} - 18 x^{3} - 211 x^{2} - 415 x - 538$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4\times C_2$ Parity: Even Determinant: 1.29_229.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $x^{2} + 49 x + 2$
Roots:
 $r_{ 1 }$ $=$ $11 + 37\cdot 53 + 49\cdot 53^{2} + 32\cdot 53^{3} + 39\cdot 53^{4} + 6\cdot 53^{5} + 34\cdot 53^{6} + 7\cdot 53^{7} + 7\cdot 53^{8} + 12\cdot 53^{9} + 23\cdot 53^{10} + 17\cdot 53^{11} +O\left(53^{ 12 }\right)$ $r_{ 2 }$ $=$ $11 a + 16 + \left(41 a + 52\right)\cdot 53 + \left(7 a + 18\right)\cdot 53^{2} + \left(38 a + 11\right)\cdot 53^{3} + \left(16 a + 30\right)\cdot 53^{4} + \left(26 a + 34\right)\cdot 53^{5} + \left(30 a + 34\right)\cdot 53^{6} + \left(7 a + 17\right)\cdot 53^{7} + \left(13 a + 16\right)\cdot 53^{8} + \left(4 a + 18\right)\cdot 53^{9} + \left(24 a + 15\right)\cdot 53^{10} + \left(26 a + 30\right)\cdot 53^{11} +O\left(53^{ 12 }\right)$ $r_{ 3 }$ $=$ $42 a + 7 + \left(11 a + 47\right)\cdot 53 + \left(45 a + 8\right)\cdot 53^{2} + \left(14 a + 50\right)\cdot 53^{3} + \left(36 a + 5\right)\cdot 53^{4} + \left(26 a + 17\right)\cdot 53^{5} + \left(22 a + 24\right)\cdot 53^{6} + \left(45 a + 17\right)\cdot 53^{7} + \left(39 a + 8\right)\cdot 53^{8} + \left(48 a + 22\right)\cdot 53^{9} + \left(28 a + 1\right)\cdot 53^{10} + \left(26 a + 6\right)\cdot 53^{11} +O\left(53^{ 12 }\right)$ $r_{ 4 }$ $=$ $33 + 6\cdot 53 + 20\cdot 53^{2} + 28\cdot 53^{3} + 22\cdot 53^{4} + 45\cdot 53^{5} + 35\cdot 53^{6} + 5\cdot 53^{7} + 46\cdot 53^{8} + 44\cdot 53^{9} + 40\cdot 53^{10} + 7\cdot 53^{11} +O\left(53^{ 12 }\right)$ $r_{ 5 }$ $=$ $2 a + 43 + \left(33 a + 48\right)\cdot 53 + \left(50 a + 51\right)\cdot 53^{2} + \left(44 a + 32\right)\cdot 53^{3} + \left(24 a + 29\right)\cdot 53^{4} + \left(50 a + 18\right)\cdot 53^{5} + \left(50 a + 44\right)\cdot 53^{6} + \left(31 a + 16\right)\cdot 53^{7} + \left(5 a + 45\right)\cdot 53^{8} + \left(24 a + 11\right)\cdot 53^{9} + \left(29 a + 45\right)\cdot 53^{10} + \left(12 a + 11\right)\cdot 53^{11} +O\left(53^{ 12 }\right)$ $r_{ 6 }$ $=$ $51 a + 51 + \left(19 a + 19\right)\cdot 53 + \left(2 a + 9\right)\cdot 53^{2} + \left(8 a + 3\right)\cdot 53^{3} + \left(28 a + 31\right)\cdot 53^{4} + \left(2 a + 36\right)\cdot 53^{5} + \left(2 a + 38\right)\cdot 53^{6} + \left(21 a + 40\right)\cdot 53^{7} + \left(47 a + 35\right)\cdot 53^{8} + \left(28 a + 49\right)\cdot 53^{9} + \left(23 a + 32\right)\cdot 53^{10} + \left(40 a + 32\right)\cdot 53^{11} +O\left(53^{ 12 }\right)$

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

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

### Character values on conjugacy classes

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