Abelian variety isogeny class 2.5.ab_k over $\F_{5}$

• Label: 2.5.ab_k
• Base field: $\F_{5}$
• 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:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$( 1 - x + 5 x^{2} )( 1 + 5 x^{2} )$
	$1 - x + 10 x^{2} - 5 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.428216853436$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$30$	$1260$	$17640$	$342720$	$9456150$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$45$	$140$	$545$	$3025$	$15930$	$78685$	$389665$	$1950620$	$9767925$

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 $\F_{5}$

The isogeny class factors as 1.5.ab $\times$ 1.5.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.5.ab : \(\Q(\sqrt{-19}) \).
• 1.5.a : \(\Q(\sqrt{-5}) \).

Endomorphism algebra over $\overline{\F}_{5}$

The base change of $A$ to $\F_{5^{2}}$ is 1.25.j $\times$ 1.25.k. The endomorphism algebra for each factor is:

• 1.25.j : \(\Q(\sqrt{-19}) \).
• 1.25.k : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.

Base change

This is a primitive isogeny class.

Twists

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

Twist	Extension degree	Common base change
2.5.b_k	$2$	2.25.t_fk