# Properties

 Label 4.7260125.12t34.a Dimension $4$ Group $C_3^2:D_4$ Conductor $7260125$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$7260125$$$$\medspace = 5^{3} \cdot 241^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.30125.1 Galois orbit size: $1$ Smallest permutation container: 12T34 Parity: even Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.2.30125.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $4$ $6$ $2$ $(1,3)(2,5)(4,6)$ $-2$ $6$ $2$ $(2,4)$ $0$ $9$ $2$ $(2,4)(5,6)$ $0$ $4$ $3$ $(1,2,4)$ $-2$ $4$ $3$ $(1,2,4)(3,5,6)$ $1$ $18$ $4$ $(1,3)(2,6,4,5)$ $0$ $12$ $6$ $(1,5,2,6,4,3)$ $1$ $12$ $6$ $(2,4)(3,5,6)$ $0$
The blue line marks the conjugacy class containing complex conjugation.