Properties

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

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Invariants

Base field:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $( 1 - 4 x + 5 x^{2} )( 1 + 3 x + 5 x^{2} )$
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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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