# Properties

 Label 2.5.ab_f 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: $1 - x + 5 x^{2} - 5 x^{3} + 25 x^{4}$ Frobenius angles: $\pm0.285445024444$, $\pm0.631178887038$ Angle rank: $2$ (numerical) Number field: 4.0.90405.2 Galois group: $D_{4}$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 25 925 15325 422725 10162000 238717525 6048984925 152763092325 3812940502225 95388173824000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 35 125 675 3250 15275 77425 391075 1952225 9767750

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is 4.0.90405.2.
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_f $2$ 2.25.j_cn