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

• Label: 2.25.an_dc
• 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:	$( 1 - 5 x )^{2}( 1 - 3 x + 25 x^{2} )$
	$1 - 13 x + 80 x^{2} - 325 x^{3} + 625 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.403013315979$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$368$	$384192$	$243309824$	$151932568320$	$95245488225008$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$617$	$15574$	$388945$	$9753133$	$244101422$	$6103491589$	$152587704865$	$3814691840518$	$95367392665577$

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.ad 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.ad : \(\Q(\sqrt{-91}) \).

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.ah_u	$2$	2.625.aj_abeu
2.25.h_u	$2$	2.625.aj_abeu
2.25.n_dc	$2$	2.625.aj_abeu
2.25.c_bj	$3$	(not in LMFDB)