# Properties

 Label 2.5.ab_e Base Field $\F_{5}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{5}$ Dimension: $2$ L-polynomial: $( 1 - 3 x + 5 x^{2} )( 1 + 2 x + 5 x^{2} )$ Frobenius angles: $\pm0.265942140215$, $\pm0.647583617650$ Angle rank: $2$ (numerical) Jacobians: 3

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

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

## Point counts

This isogeny class contains the Jacobians of 3 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 24 864 14976 432000 10211064 239376384 6068563896 152549568000 3808226807424 95392127486304

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 33 122 689 3265 15318 77677 390529 1949810 9768153

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ad $\times$ 1.5.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{5}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.af_q $2$ 2.25.h_ce 2.5.b_e $2$ 2.25.h_ce 2.5.f_q $2$ 2.25.h_ce
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.af_q $2$ 2.25.h_ce 2.5.b_e $2$ 2.25.h_ce 2.5.f_q $2$ 2.25.h_ce 2.5.ah_w $4$ 2.625.cl_cwm 2.5.ab_ac $4$ 2.625.cl_cwm 2.5.b_ac $4$ 2.625.cl_cwm 2.5.h_w $4$ 2.625.cl_cwm