# Properties

 Label 4.2312000.6t10.d.a Dimension 4 Group $C_3^2:C_4$ Conductor $2^{6} \cdot 5^{3} \cdot 17^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:C_4$ Conductor: $2312000= 2^{6} \cdot 5^{3} \cdot 17^{2}$ Artin number field: Splitting field of 6.2.11560000.3 defined by $f= x^{6} - 2 x^{5} + 4 x^{4} + 4 x^{3} - 6 x^{2} + 18 x + 1$ 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 6.2.11560000.3

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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