Abelian variety isogeny class 2.25.ar_es over $\F_{5^{2}}$

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

Invariants

Base field:	$\F_{5^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x + 25 x^{2} )( 1 - 8 x + 25 x^{2} )$
	$1 - 17 x + 122 x^{2} - 425 x^{3} + 625 x^{4}$
Frobenius angles:	$\pm0.143566293129$, $\pm0.204832764699$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	3

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$306$	$364140$	$244698408$	$153113587200$	$95466919164066$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$581$	$15660$	$391969$	$9775809$	$244192466$	$6103704105$	$152588258689$	$3814696195884$	$95367416254901$

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.aj $\times$ 1.25.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.25.aj : \(\Q(\sqrt{-19}) \).
• 1.25.ai : \(\Q(\sqrt{-1}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.25.ab_aw	$2$	2.625.abt_cmu
2.25.b_aw	$2$	2.625.abt_cmu
2.25.r_es	$2$	2.625.abt_cmu