# Properties

 Label 5.13e4_19e4.6t15.2c1 Dimension 5 Group $A_6$ Conductor $13^{4} \cdot 19^{4}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $A_6$ Conductor: $3722098081= 13^{4} \cdot 19^{4}$ Artin number field: Splitting field of $f= x^{6} - 4 x^{4} - 15 x^{3} - 15 x^{2} - 8 x + 4$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $A_6$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $x^{2} + 70 x + 5$
Roots:
 $r_{ 1 }$ $=$ $6 a + 33 + \left(70 a + 4\right)\cdot 73 + \left(62 a + 45\right)\cdot 73^{2} + \left(59 a + 29\right)\cdot 73^{3} + \left(25 a + 36\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 2 }$ $=$ $24 + 44\cdot 73 + 38\cdot 73^{2} + 28\cdot 73^{3} + 22\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 3 }$ $=$ $48 + 50\cdot 73 + 26\cdot 73^{2} + 22\cdot 73^{3} + 41\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 4 }$ $=$ $36 a + 14 + \left(58 a + 68\right)\cdot 73 + \left(71 a + 39\right)\cdot 73^{2} + \left(36 a + 49\right)\cdot 73^{3} + \left(22 a + 53\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 5 }$ $=$ $37 a + 49 + \left(14 a + 61\right)\cdot 73 + \left(a + 50\right)\cdot 73^{2} + \left(36 a + 15\right)\cdot 73^{3} + \left(50 a + 11\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 6 }$ $=$ $67 a + 51 + \left(2 a + 62\right)\cdot 73 + \left(10 a + 17\right)\cdot 73^{2} + 13 a\cdot 73^{3} + \left(47 a + 54\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $5$ $45$ $2$ $(1,2)(3,4)$ $1$ $40$ $3$ $(1,2,3)(4,5,6)$ $2$ $40$ $3$ $(1,2,3)$ $-1$ $90$ $4$ $(1,2,3,4)(5,6)$ $-1$ $72$ $5$ $(1,2,3,4,5)$ $0$ $72$ $5$ $(1,3,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.