# Properties

 Label 4.2e2_7e2_13e2_29.6t13.1c1 Dimension 4 Group $C_3^2:D_4$ Conductor $2^{2} \cdot 7^{2} \cdot 13^{2} \cdot 29$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $960596= 2^{2} \cdot 7^{2} \cdot 13^{2} \cdot 29$ Artin number field: Splitting field of $f= x^{6} - 10 x^{4} - 4 x^{3} - 4 x^{2} + 20 x + 4$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.29.2t1.1c1

## Galois action

### Roots of defining polynomial

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

### 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 $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $2$ $6$ $2$ $(1,4)$ $0$ $9$ $2$ $(1,4)(2,3)$ $0$ $4$ $3$ $(1,4,5)(2,3,6)$ $1$ $4$ $3$ $(2,3,6)$ $-2$ $18$ $4$ $(1,3,4,2)(5,6)$ $0$ $12$ $6$ $(1,2,4,3,5,6)$ $-1$ $12$ $6$ $(1,4)(2,3,6)$ $0$
The blue line marks the conjugacy class containing complex conjugation.