# Properties

 Label 4.2e4_5_149.6t13.2c1 Dimension 4 Group $C_3^2:D_4$ Conductor $2^{4} \cdot 5 \cdot 149$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $11920= 2^{4} \cdot 5 \cdot 149$ Artin number field: Splitting field of $f= x^{6} + x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.5_149.2t1.1c1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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