# Properties

 Label 4.1514670125.6t10.a.a Dimension 4 Group $C_3^2:C_4$ Conductor $5^{3} \cdot 59^{4}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:C_4$ Conductor: $1514670125= 5^{3} \cdot 59^{4}$ Artin number field: Splitting field of 6.2.2175625.1 defined by $f= x^{6} - x^{5} + x^{4} - 7 x^{3} - 6 x^{2} - 6 x - 1$ 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 6.2.2175625.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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