Properties

Label 2.25.ao_dm
Base field $\F_{5^{2}}$
Dimension $2$
$p$-rank $1$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{5^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x )^{2}( 1 - 4 x + 25 x^{2} )$
  $1 - 14 x + 90 x^{2} - 350 x^{3} + 625 x^{4}$
Frobenius angles:  $0$, $0$, $\pm0.369010119566$
Angle rank:  $1$ (numerical)
Jacobians:  $0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $352$ $380160$ $243894112$ $152136990720$ $95252505216352$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $12$ $610$ $15612$ $389470$ $9753852$ $244084930$ $6103399692$ $152587881790$ $3814695441132$ $95367401126050$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{2}}$.

Endomorphism algebra over $\F_{5^{2}}$
The isogeny class factors as 1.25.ak $\times$ 1.25.ae 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
2.25.ag_k$2$2.625.aq_ari
2.25.g_k$2$2.625.aq_ari
2.25.o_dm$2$2.625.aq_ari
2.25.b_be$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.25.ag_k$2$2.625.aq_ari
2.25.g_k$2$2.625.aq_ari
2.25.o_dm$2$2.625.aq_ari
2.25.b_be$3$(not in LMFDB)
2.25.aj_cs$6$(not in LMFDB)
2.25.ab_be$6$(not in LMFDB)
2.25.j_cs$6$(not in LMFDB)