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

• Label: 2.25.ap_ec
• 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:	$( 1 - 8 x + 25 x^{2} )( 1 - 7 x + 25 x^{2} )$
	$1 - 15 x + 106 x^{2} - 375 x^{3} + 625 x^{4}$
Frobenius angles:	$\pm0.204832764699$, $\pm0.253183311107$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$342$	$383724$	$248406912$	$153489600000$	$95469283953702$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$613$	$15896$	$392929$	$9776051$	$244162258$	$6103429931$	$152586782209$	$3814691319416$	$95367414284053$

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.ai $\times$ 1.25.ah 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.ai : \(\Q(\sqrt{-1}) \).
• 1.25.ah : \(\Q(\sqrt{-51}) \).

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_ag	$2$	2.625.an_bvo
2.25.b_ag	$2$	2.625.an_bvo
2.25.p_ec	$2$	2.625.an_bvo