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

• Label: 2.25.as_fa
• 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

Invariants

Base field:	$\F_{5^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x )^{2}( 1 - 8 x + 25 x^{2} )$
	$1 - 18 x + 130 x^{2} - 450 x^{3} + 625 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.204832764699$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$288$	$352512$	$241618464$	$152510791680$	$95367236440608$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$562$	$15464$	$390430$	$9765608$	$244132882$	$6103391624$	$152586779710$	$3814689915848$	$95367392802802$

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.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.ak : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
• 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.ac_abe	$2$	2.625.acm_cxa
2.25.c_abe	$2$	2.625.acm_cxa
2.25.s_fa	$2$	2.625.acm_cxa
2.25.ad_k	$3$	(not in LMFDB)