# Properties

 Label 2.5.ab_a Base Field $\F_{5}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{5}$ Dimension: $2$ L-polynomial: $1 - x - 5 x^{3} + 25 x^{4}$ Frobenius angles: $\pm0.189652078905$, $\pm0.706462326177$ Angle rank: $2$ (numerical) Number field: 4.0.8405.1 Galois group: $D_{4}$ Jacobians: 4

This isogeny class is simple and geometrically simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

• $y^2=2x^5+4x^2+1$
• $y^2=x^5+4x^2+2$
• $y^2=4x^6+2x^5+3x^3+4x+2$
• $y^2=2x^6+2x^4+4x^3+x+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 20 640 13760 442880 10080100 244817920 6169378340 152338319360 3809046956480 95369358275200

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 25 110 705 3225 15670 78965 389985 1950230 9765825

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is 4.0.8405.1.
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 2.5.b_a $2$ 2.25.ab_bo