Properties

Label 2.5.ae_k
Base Field $\F_{5}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

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

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 12 720 15372 368640 10009452 251793360 6147109932 152510791680 3817505860812 95433118203600

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 30 122 590 3202 16110 78682 390430 1954562 9772350

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The isogeny class factors as 1.5.ae $\times$ 1.5.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag $\times$ 1.25.k. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{5^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.e_k$2$2.25.e_ak
2.5.ac_k$4$2.625.abk_ve
2.5.c_k$4$2.625.abk_ve