# Properties

 Label 2.29e2_229.6t3.2c1 Dimension 2 Group $D_{6}$ Conductor $29^{2} \cdot 229$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $192589= 29^{2} \cdot 229$ Artin number field: Splitting field of $f= x^{6} - 176 x^{4} - 65 x^{3} + 7744 x^{2} + 5720 x - 80296$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{6}$ Parity: Even Determinant: 1.229.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $x^{2} + 29 x + 3$
Roots:
 $r_{ 1 }$ $=$ $29 + 26\cdot 31 + 3\cdot 31^{2} + 10\cdot 31^{3} + 4\cdot 31^{4} + 15\cdot 31^{5} + 26\cdot 31^{6} + 12\cdot 31^{7} + 2\cdot 31^{8} + 15\cdot 31^{9} +O\left(31^{ 10 }\right)$ $r_{ 2 }$ $=$ $17 a + 8 + \left(28 a + 24\right)\cdot 31 + \left(9 a + 11\right)\cdot 31^{2} + \left(6 a + 15\right)\cdot 31^{3} + \left(22 a + 13\right)\cdot 31^{4} + \left(28 a + 24\right)\cdot 31^{5} + 28\cdot 31^{6} + \left(25 a + 29\right)\cdot 31^{7} + \left(29 a + 21\right)\cdot 31^{8} + \left(28 a + 27\right)\cdot 31^{9} +O\left(31^{ 10 }\right)$ $r_{ 3 }$ $=$ $29 a + 3 + \left(10 a + 21\right)\cdot 31 + \left(a + 17\right)\cdot 31^{2} + \left(29 a + 28\right)\cdot 31^{3} + \left(23 a + 3\right)\cdot 31^{4} + 19 a\cdot 31^{5} + \left(19 a + 8\right)\cdot 31^{6} + \left(12 a + 6\right)\cdot 31^{7} + 20\cdot 31^{8} + \left(27 a + 27\right)\cdot 31^{9} +O\left(31^{ 10 }\right)$ $r_{ 4 }$ $=$ $2 a + 30 + \left(20 a + 13\right)\cdot 31 + \left(29 a + 9\right)\cdot 31^{2} + \left(a + 23\right)\cdot 31^{3} + \left(7 a + 22\right)\cdot 31^{4} + \left(11 a + 15\right)\cdot 31^{5} + \left(11 a + 27\right)\cdot 31^{6} + \left(18 a + 11\right)\cdot 31^{7} + \left(30 a + 8\right)\cdot 31^{8} + \left(3 a + 19\right)\cdot 31^{9} +O\left(31^{ 10 }\right)$ $r_{ 5 }$ $=$ $12 + 4\cdot 31 + 16\cdot 31^{2} + 28\cdot 31^{3} + 27\cdot 31^{4} + 8\cdot 31^{5} + 15\cdot 31^{7} + 14\cdot 31^{8} + 9\cdot 31^{9} +O\left(31^{ 10 }\right)$ $r_{ 6 }$ $=$ $14 a + 11 + \left(2 a + 2\right)\cdot 31 + \left(21 a + 3\right)\cdot 31^{2} + \left(24 a + 18\right)\cdot 31^{3} + \left(8 a + 20\right)\cdot 31^{4} + \left(2 a + 28\right)\cdot 31^{5} + \left(30 a + 1\right)\cdot 31^{6} + \left(5 a + 17\right)\cdot 31^{7} + \left(a + 25\right)\cdot 31^{8} + \left(2 a + 24\right)\cdot 31^{9} +O\left(31^{ 10 }\right)$

### 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.