# Properties

 Label 4.5e2_421e3.12t36.1c1 Dimension 4 Group $C_3^2:D_4$ Conductor $5^{2} \cdot 421^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $1865461525= 5^{2} \cdot 421^{3}$ Artin number field: Splitting field of $f= x^{6} - 2 x^{4} - x^{3} + x^{2} + x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 12T36 Parity: Even Determinant: 1.421.2t1.1c1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,5)$ $-2$ $9$ $2$ $(3,5)(4,6)$ $0$ $4$ $3$ $(1,4,6)$ $1$ $4$ $3$ $(1,4,6)(2,3,5)$ $-2$ $18$ $4$ $(1,2)(3,6,5,4)$ $0$ $12$ $6$ $(1,3,4,5,6,2)$ $0$ $12$ $6$ $(1,4,6)(3,5)$ $1$
The blue line marks the conjugacy class containing complex conjugation.