# Properties

 Label 3.5.ah_y_ach Base Field $\F_{5}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{5}$ Dimension: $3$ L-polynomial: $1 - 7 x + 24 x^{2} - 59 x^{3} + 120 x^{4} - 175 x^{5} + 125 x^{6}$ Frobenius angles: $\pm0.0749012311065$, $\pm0.225515375241$, $\pm0.553262127050$ Angle rank: $3$ (numerical) Number field: 6.0.29215664.1 Galois group: $S_4\times C_2$ Jacobians: 0

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1, 1, 1]$

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 29 14819 1730372 235814747 31945643129 3865810242224 473141662652821 59463876907244187 7456146350377582004 931267264837529756339

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 25 110 605 3269 15832 77517 389701 1954586 9765045

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is 6.0.29215664.1.
All geometric endomorphisms are defined over $\F_{5}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.5.h_y_ch $2$ 3.25.ab_ak_db