# Properties

 Label 4.117128000.6t10.a.a Dimension 4 Group $C_3^2:C_4$ Conductor $2^{6} \cdot 5^{3} \cdot 11^{4}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:C_4$ Conductor: $117128000= 2^{6} \cdot 5^{3} \cdot 11^{4}$ Artin number field: Splitting field of 6.2.4840000.2 defined by $f= x^{6} - x^{5} + x^{4} - 12 x^{3} - 21 x^{2} - 11 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:C_4$ Parity: Even Determinant: 1.5.2t1.a.a Projective image: $C_3:S_3.C_2$ Projective field: Galois closure of 6.2.4840000.2

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $x^{2} + 60 x + 2$
Roots:
 $r_{ 1 }$ $=$ $2 a + 32 + \left(41 a + 43\right)\cdot 61 + \left(2 a + 26\right)\cdot 61^{2} + \left(47 a + 25\right)\cdot 61^{3} + \left(50 a + 25\right)\cdot 61^{4} + \left(8 a + 11\right)\cdot 61^{5} + \left(7 a + 15\right)\cdot 61^{6} + \left(29 a + 13\right)\cdot 61^{7} + \left(28 a + 6\right)\cdot 61^{8} + \left(60 a + 25\right)\cdot 61^{9} + \left(23 a + 34\right)\cdot 61^{10} + \left(58 a + 42\right)\cdot 61^{11} + \left(56 a + 25\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ $r_{ 2 }$ $=$ $59 a + 34 + \left(19 a + 21\right)\cdot 61 + \left(58 a + 49\right)\cdot 61^{2} + \left(13 a + 8\right)\cdot 61^{3} + \left(10 a + 29\right)\cdot 61^{4} + \left(52 a + 30\right)\cdot 61^{5} + \left(53 a + 13\right)\cdot 61^{6} + \left(31 a + 35\right)\cdot 61^{7} + \left(32 a + 5\right)\cdot 61^{8} + 57\cdot 61^{9} + \left(37 a + 58\right)\cdot 61^{10} + \left(2 a + 15\right)\cdot 61^{11} + \left(4 a + 24\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ $r_{ 3 }$ $=$ $58 + 13\cdot 61 + 14\cdot 61^{2} + 34\cdot 61^{3} + 28\cdot 61^{4} + 6\cdot 61^{5} + 16\cdot 61^{6} + 55\cdot 61^{7} + 2\cdot 61^{8} + 44\cdot 61^{9} + 6\cdot 61^{10} + 29\cdot 61^{11} + 50\cdot 61^{12} +O\left(61^{ 13 }\right)$ $r_{ 4 }$ $=$ $45 a + 49 + \left(60 a + 28\right)\cdot 61 + 44\cdot 61^{2} + \left(16 a + 14\right)\cdot 61^{3} + \left(54 a + 27\right)\cdot 61^{4} + \left(19 a + 45\right)\cdot 61^{5} + \left(a + 28\right)\cdot 61^{6} + \left(46 a + 33\right)\cdot 61^{7} + 25\cdot 61^{8} + \left(58 a + 7\right)\cdot 61^{9} + \left(47 a + 35\right)\cdot 61^{10} + \left(38 a + 55\right)\cdot 61^{11} + \left(25 a + 10\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$ $r_{ 5 }$ $=$ $39 + 30\cdot 61 + 2\cdot 61^{2} + 9\cdot 61^{3} + 7\cdot 61^{4} + 17\cdot 61^{5} + 38\cdot 61^{6} + 28\cdot 61^{7} + 40\cdot 61^{8} + 45\cdot 61^{9} + 22\cdot 61^{10} + 54\cdot 61^{11} + 12\cdot 61^{12} +O\left(61^{ 13 }\right)$ $r_{ 6 }$ $=$ $16 a + 33 + 44\cdot 61 + \left(60 a + 45\right)\cdot 61^{2} + \left(44 a + 29\right)\cdot 61^{3} + \left(6 a + 4\right)\cdot 61^{4} + \left(41 a + 11\right)\cdot 61^{5} + \left(59 a + 10\right)\cdot 61^{6} + \left(14 a + 17\right)\cdot 61^{7} + \left(60 a + 41\right)\cdot 61^{8} + \left(2 a + 3\right)\cdot 61^{9} + \left(13 a + 25\right)\cdot 61^{10} + \left(22 a + 46\right)\cdot 61^{11} + \left(35 a + 58\right)\cdot 61^{12} +O\left(61^{ 13 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $9$ $2$ $(1,2)(3,4)$ $0$ $4$ $3$ $(1,2,5)$ $-2$ $4$ $3$ $(1,2,5)(3,4,6)$ $1$ $9$ $4$ $(1,4,2,3)(5,6)$ $0$ $9$ $4$ $(1,3,2,4)(5,6)$ $0$
The blue line marks the conjugacy class containing complex conjugation.