Properties

Label 2.5.af_o
Base field $\F_{5}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $( 1 - 4 x + 5 x^{2} )( 1 - x + 5 x^{2} )$
  $1 - 5 x + 14 x^{2} - 25 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.147583617650$, $\pm0.428216853436$
Angle rank:  $2$ (numerical)
Jacobians:  $1$
Isomorphism classes:  3

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $10$ $700$ $17080$ $380800$ $9686050$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $29$ $136$ $609$ $3101$ $15914$ $79241$ $391969$ $1952056$ $9762149$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5}$.

Endomorphism algebra over $\F_{5}$
The isogeny class factors as 1.5.ae $\times$ 1.5.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.5.ad_g$2$2.25.d_ae
2.5.d_g$2$2.25.d_ae
2.5.f_o$2$2.25.d_ae

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.5.ad_g$2$2.25.d_ae
2.5.d_g$2$2.25.d_ae
2.5.f_o$2$2.25.d_ae
2.5.ad_m$4$2.625.ar_bfk
2.5.ab_i$4$2.625.ar_bfk
2.5.b_i$4$2.625.ar_bfk
2.5.d_m$4$2.625.ar_bfk