# Properties

 Label 4.14225.6t13.a Dimension $4$ Group $C_3^2:D_4$ Conductor $14225$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$14225$$$$\medspace = 5^{2} \cdot 569$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.71125.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: even Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.2.71125.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $x^{2} + 60 x + 2$
Roots:
 $r_{ 1 }$ $=$ $37 a + 9 + \left(56 a + 19\right)\cdot 61 + \left(58 a + 10\right)\cdot 61^{2} + \left(53 a + 23\right)\cdot 61^{3} + \left(57 a + 48\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 2 }$ $=$ $27 + 54\cdot 61 + 32\cdot 61^{2} + 22\cdot 61^{3} + 55\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 3 }$ $=$ $11 a + 12 + \left(21 a + 59\right)\cdot 61 + \left(19 a + 14\right)\cdot 61^{2} + \left(58 a + 30\right)\cdot 61^{3} + \left(25 a + 49\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 4 }$ $=$ $24 a + 46 + \left(4 a + 38\right)\cdot 61 + \left(2 a + 12\right)\cdot 61^{2} + \left(7 a + 18\right)\cdot 61^{3} + \left(3 a + 52\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 5 }$ $=$ $7 + 3\cdot 61 + 38\cdot 61^{2} + 19\cdot 61^{3} + 21\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 6 }$ $=$ $50 a + 23 + \left(39 a + 8\right)\cdot 61 + \left(41 a + 13\right)\cdot 61^{2} + \left(2 a + 8\right)\cdot 61^{3} + \left(35 a + 17\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$

### 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 values $c1$ $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,6)$ $2$ $9$ $2$ $(3,6)(4,5)$ $0$ $4$ $3$ $(1,4,5)$ $1$ $4$ $3$ $(1,4,5)(2,3,6)$ $-2$ $18$ $4$ $(1,2)(3,5,6,4)$ $0$ $12$ $6$ $(1,3,4,6,5,2)$ $0$ $12$ $6$ $(1,4,5)(3,6)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.