# Properties

 Label 2.5.ab_ac 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: $( 1 - 4 x + 5 x^{2} )( 1 + 3 x + 5 x^{2} )$ Frobenius angles: $\pm0.147583617650$, $\pm0.734057859785$ Angle rank: $2$ (numerical) Jacobians: 2

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 2 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 18 540 13176 432000 9826938 246654720 6187001706 152549568000 3819300148728 95401388144700

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 21 104 689 3145 15786 79189 390529 1955480 9769101

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae $\times$ 1.5.d 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.ah_w $2$ 2.25.af_bs 2.5.b_ac $2$ 2.25.af_bs 2.5.h_w $2$ 2.25.af_bs
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.ah_w $2$ 2.25.af_bs 2.5.b_ac $2$ 2.25.af_bs 2.5.h_w $2$ 2.25.af_bs 2.5.af_q $4$ 2.625.cl_cwm 2.5.ab_e $4$ 2.625.cl_cwm 2.5.b_e $4$ 2.625.cl_cwm 2.5.f_q $4$ 2.625.cl_cwm