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

• Label: 2.5.ac_j
• Base field: $\F_{5}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$1 - 2 x + 9 x^{2} - 10 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.318486791133$, $\pm0.529524486268$
Angle rank:	$2$ (numerical)
Number field:	4.0.17984.1
Galois group:	$D_{4}$
Jacobians:	$1$
Isomorphism classes:	1

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$23$	$1081$	$17756$	$381593$	$9826543$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$40$	$142$	$612$	$3144$	$15598$	$77368$	$389892$	$1957750$	$9774600$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=2x^6+x^5+3x^4+3x^2+2x+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5}$.

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is 4.0.17984.1.

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.c_j	$2$	2.25.o_dn