Properties

Label 2.625.ado_eyw
Base Field $\F_{5^{4}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 25 x )^{2}( 1 - 42 x + 625 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.182554890994$
Angle rank:  $1$ (numerical)

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 336384 151900250112 59598153907020288 23283024052177838407680 9094946948231806682997318144 3552713677505038927677090554437632 1387778780618887762592355702943634416128 542101086236527609911526184116318549793832960 211758236813415848684579493959383319153333502779904 82718061255299552107928778299666734512092522899643031552

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 534 388862 244114038 152587626430 95367430911894 59604644753656382 37252902980255509878 23283064365119618051710 14551915228355909186373654 9094947017728928844467250302

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The isogeny class factors as 1.625.aby $\times$ 1.625.abq 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 $\F_{5^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.625.ai_abgs$2$(not in LMFDB)
2.625.i_abgs$2$(not in LMFDB)
2.625.do_eyw$2$(not in LMFDB)
2.625.ar_hs$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.625.ai_abgs$2$(not in LMFDB)
2.625.i_abgs$2$(not in LMFDB)
2.625.do_eyw$2$(not in LMFDB)
2.625.ar_hs$3$(not in LMFDB)
2.625.acp_dkm$6$(not in LMFDB)
2.625.r_hs$6$(not in LMFDB)
2.625.cp_dkm$6$(not in LMFDB)